HEAT
CHAPTER – 11
HEAT
DEFINITION
Total Kinetic energy of a body is known as HEAT.
OR
Transfer of energy from a hot body to a cold one is termed as Heat.
Heat is measured by using an measurement centimeter.
UNITS
Since heat is a force of energy therefore its unit is Joule (J).
TEMPERATURE
DEFINITION
The average kinetic energy of a body is known as Temperature.
OR
The quantitative determination of degree of hotness may be termed as Temperature.
SCALES OF TEMPERATURE
There are three main scales of temperature.
1. Celsius Scale
2. Fahrenheit Scale
3. Kelvin Scale
Celsius and Fahrenheit scales are also known as Scales of Graduation.
1. Celsius Scale
The melting point of ice and boiling point of water at standard pressure (76cm of Hg) taken to be two fixed points. On the Celsius (centigrade) scale the interval between these two fixed points is divided into hundred equal parts. Each part thus represents one degree Celsius (1°C). This scale was suggested by Celsius in 1742.
Mathematically,
°C = K – 273
OR
°C = 5/9 (°F – 32)
2. Fahrenheit Scale
The melting point of ice and boiling of water at standard pressure (76cm of Hg) are taken to be two fixed points. On Fahrenheit scale the lower fixed point is marked 32 and upper fixed point 212. The interval between them is equally divided into 180 parts. Each part represents one degree Fahrenheit (1°F).
Mathematically,
°F = 9/5 (°C + 32)
3. Kelvin Scale
The lowest temperature on Kelvin Scale is -273°C. Thus 0° on Celsius scale will be 273 on Kelvin scale written as 273K and 100 on Celsius scale will be 373K. The size of Celsius and Kelvin scales are same.
Mathematically,
K = °C + 273
THERMAL EQUILIBRIUM
Heat flows from hot body to cold body till the temperature of the bodies becomes same, then they are said to be in Thermal Equilibrium.
THERMAL EXPANSION
DEFINITION
The phenomenon due to which solid experience a change in its length, volume or area on heating is known as Thermal Expansion.
Explanation
If we supply some amount of heat to any substance then size or shape of the substance will increase. This increment is known as Thermal Expansion. Thermal expansion is due to the increment of the amplitudes of the molecules.
Types of Thermal Expansion
There are three types of Thermal Expansion.
1. Linear Expansion
2. Superficial Expansion
3. Volumetric Expansion.
1. Linear Expansion.
If we supply some amount of heat to any rod, then the length of the rod, then the length of the rod will increase. Such increment is known as Linear Expansion.
2. Superficial Expansion.
If we apply some amount of heat to any square or rectangle then area of the square or rectangle will increase. Such increment is known as Superficial Expansion.
3. Volumetric Expansion.
If we apply some amount of heat to any cube, then the volume of the cube will increase. Such increment is known as Volumetric Expansion.
COEFFICIENT OF LINEAR EXPANSION
CONSIDERATION
Let Lo be the initial length of rod at t1 °C. If we increase the temperature from t1 °C to t2 °C, then length of the rod will increase. This increment in length is denoted by ΔL. The increment in length depends upon the following two factors.
1. Original Length (Lo)
2. Difference in temperature Δt
Derivation
The increment in length is directly proportional to the original length and temperature difference.
Mathematically,
ΔL ∞ Lo —– (I)
ΔL ∞ Δt —– (II)
Combining eq (I) and (II), we get
ΔL ∞ LoΔt
=> ΔL = ∞LoΔt
Where α is the constant of proportionality and it is known as coefficient of Linear Expansion. It is defined as,
It is the increment in length per unit length per degree rise in temperature.
Its unit is 1/°C or °C. If Lt is the total length, then
Lt = Lo + ΔL
=> Lt = Lo + αLoΔt
=> Lt = Lo (1 + αΔt)
COEFFICIENT OF VOLUMETRIC EXPANSION
Consideration
Let Vo be the initial length of rod at t1 °C. If we increase the temperature from t1°C to t2°C then length of the rod will increase. This increment in length is denoted by ΔV. The increment in length depends upon the following two factors.
3. Original Volume (Lo)
4. Difference in temperature Δt
Derivation
The increment in volume is directly proportional to the original volume of temperature difference.
Mathematically,
ΔV ∞ Vo —- (I)
ΔV ∞ Δt —- (II)
Combining eq (I) and (II), we get,
ΔV ∞ Vo Δt
=> ΔV = βVoΔt
Where β is the constant of proportionality and it is known as coefficient of Volumetric Expansion. It is defined as
It is the increment in volume per unit volume per degree rise in temperature.
Its unit is 1/°C or °C-1. If Vt is the total volume then
Vt = Vo + ΔV
=> Vt = Vo + αβVo Δt
=> Vt = Vo (1 + βΔt)
State and Explain Boyle’s Law and Charle’s Law.
INTRODUCTION
Gas Laws are the laws, which give relationship between Pressure, Volume, temperature and mass of the gas. There are two gas laws.
1. Boyle’s Law
2. Charle’s Law
BOYLE’S LAW
Statement 1
According to first statement of Boyle’s Law:
Volume of the known mass of gas is inversely proportional to the pressure, if temperature is kept constant.
Mathematical Form
Mathematically,
V ∞ 1/P
=> V = K 1/P
=> PV = K (Constant)
P1V1 = P2V2 = … = K
=> P1V1 = P2V2
The above equation is mathematical form of Boyle’s Law.
Statement II
According to second statement of Boyle’s Law.
The product of the pressure and volume of the known mass of the gas remain constant if the temperature is kept constant.
Statement III
According to third statement of Boyle’s Law.
The product of pressure and volume of a gas is directly proportional to the mass of a gas, provided that temperature is kept constant.
Mathematical Form
Mathematically,
PV ∞ m
=> PV = Km
=> PV/m = K
=> P1V1/m1 = P2V2/m2
Limitations of Boyle’s Law
Boyle’s Law does not hold good at high pressure, because at high pressure gases convert into liquid or solid.
Graphical Representation
The graph between pressure and volume is a curved line, which shows that volume and pressure are inversely proportional to each other.
CHARLE’S LAW
Statement I
According to first statement of Charle’s Law.
Volume of known mass of gas is directly proportional to the absolute temperature, if then pressure is kept constant.
Mathematical Form
Mathematically,
V ∞ T
=> V = KT
=> V/T = K
OR
=> V1/T1 = V2/T2
The above equation is mathematical form of Charles Law.
Statement II
According to second statement of Charles Law.
The ratio between volume and temperature of the known mass of a gas is always constant, if pressure is kept constant.
Limitations of the Law
This law does not hold good at low temperature because at low temperature gases convert into liquid or solid.
GENERAL GAS EQUATION
It is the combination of Boyle’s law, Charle’s Law and Avogadro’s Law. According to Boyle’s Law.
V ∞ 1/P —- (I)
According to Charle’s Law
V ∞ T —- (II)
According to Avogadro’s Law
V ∞ n —- (III)
Combining eq (I), eq (II) and eq (III)
V α nT/P
=> V = RnT/P
=> PV = RnT —- (A)
Where R is the universal gas constant, We Know that
R = R/NA
=> R = KNA
Where K is the Boltzman constant, Its value is
K = 1.38 x 10(-23) J/K
Substituting the value of R in eq (A)
=> PV = nKNAT
=> PV = nNAKT
But nNA = N1 (Total number of molecules), therefore,
PV = NtKT
=> P = Nt/V KT
Since Nt/V = N (Total Number of molecules in a given volume), therefore,
P = NKT
The above equation is other form of General Gas Equation.
Qs. What are the basic postulates of Kinetic Molecular Theory pf Gases?
INTRODUCTION
The properties of matter in bulk can however be predicted on molecular basis by a theory known as Kinetic Molecular theory of gases. The characteristic of this theory are described by some fundamental assumptions, which explained below:
BASIC POSTULATES OF KINETIC MOLECULAR THEORY OF GASES
1. Composition
All gases are composed of small, spherical solid particle called molecules.
2. Dimension of Molecules
The dimensions of the molecules is compared to the separation between the molecule is very small.
3. Number of Molecules
At standard condition, there are 3 x 10(23) molecules in a cubic meter.
4. Pressure of Gas
Gas molecules collide with each other as well as with the wall of the container and exert force on the walls of the container. This force per unit are is known as Pressure.
5. Collision Between the Molecules
The collision between the molecules is elastic in which momentum and Kinetic energy remains constant.
7. Kinetic Energy of Molecules
If we increase the temperature of gas molecules, then K.E will also increase. It means that average kinetic energy of the gas molecules is directly proportional to the absolute temperature.
8. Forces Of Interraction
There is no force of attraction or repulsion between the molecules.
9. Law of Mechanics
Newtonian mechanics is applicable to the motion of molecules.
THERMODYNAMICS
DEFINITIONS
The branch of Physics that deals with the conversion of heat energy into mechanical energy or work or transformation of work into heat energy is known as Thermodynamics.
Laws of Thermodynamics
There are two laws of thermodynamics.
1. First Law of Thermodynamics
2. Second Law of Thermodynamics
State and explain first law of Thermodynamics. What are the application of first law of Thermodynamics?
FIRST LAW OF THERMODYNAMICS
First Statement
Whenever heat energy is converted into work or work is transformed into heat energy, the total amount of heat energy is directly proportional to the total amount of work done.
Mathematical Expression
Mathematically,
Q ∞ W
=> Q = JW
Where J is the mechanical equivalent of heat or joules constant. Its value is 4.2 joules.
Second Statement
If ΔQ is the amount of heat supplied to any system, then this heat will be utilized to increase the internal energy of the system in the work done in order to move the piston.
Mathematical Expression
Mathematically,
ΔQ – Au + Δw
The above equation is the mathematical form of first law of thermodynamics.
Where
Δu = Internal energy of the system.
Δw = Amount of work done.
ΔQ will be positive when heat is supplied to the system and it is negative when heat is rejected by the system.
Δw will be positive when work is done by the system and it will be negative when work is done on the system.
Third Statement
For a cyclic process, the heat energy supplied to a system and work done on the system is equal to the sum of heat energy rejected by the system.
Mathematical Expression
Mathematically,
Q(IN) + W(IN) = Q(OUT) + W(OUT)
Q(IN) – Q(OUT) = W(OUT) + W(IN)
ΔQ = ΔW
{dQ = {dW
{Shows cyclic process
Fourth Statement
For a system and surrounding the total amount of heat energy remains constant
APPLICATIONS OF THE LAW
There are four applications of first law of Thermodynamics.
1. Isometric or Isocohric Process.
2. Isobaric Process
3. Isothermal Process
4. Adiabatic Process
1. Isometric or Isocohric Process
The process in which volume of the system remains constant is known as Isometric Process.
In this process all supplied amount of heat is utilized to increase the internal energy of the system.
Mathematical Form
In this process first law of thermodynamics take the following form.
ΔQ = Δu + ΔW
But,
ΔW = 0
=> ΔQ = Δu = 0
=> ΔQ = Δu
2. Isobaric Process
The process in which pressure is kept constant is known as isobaric process.
In this process, all supplied amount of heat is utilized for the following two functions.
i. To increase the internal energy of the system.
ii. In work done in order to move the piston upward.
3. Isothermal Process
A process in which temperature is kept constant is known as Isothermal Process.
There are two parts of isothermal process.
i. Isothermal Expansion
ii. Isothermal Compression
i. Isothermal Expansion
In this process cyclinder is placed on a source and piston is allowed to move upward. When we do so temperature and pressure of the working substance will decrease while volume will increase. In order to keep the temperature constant, we have to supply required amount of heat from source to cylinder.
Since in this expansion, temperature is constant therefore it is known as Isothermal Expansion.
ii. Isothermal Compression
In this process, cylinder is placed on a sink and piston is allowed to move downward. When we do so temperature and pressure of working substance will increase while volume will decrease. In order to maintain the temperature, we have to reject required amount of heat from cylinder to the sink.
Since in this compression, temperature is kept constant therefore it is known as isothermal compression.
SECOND LAW OF THERMODYNAMICS
Introduction
It is inherit tendency of heat that it always flows from hot reservoir to cold reservoir. Rather than to flow in both the directions with equal probability. On the basis of this tendency of heat a law was proposed that is known as Second Law of Thermodynamics.
Statement
It is impossible to construct a process which reserves the natural tendency of heat.
This law is also known as Law of heat and can also be stated as
Efficiency of heat engine is always less than unity.
Explanation
Many statements of this law has been proposed to cover similar but different point of vies in which two are given below.
1. Lord Kelvin Statement
2. Clausius Statement
1. Lord Kelvin Statement
According to this statement,
It is impossible to construct a heat engine which extract all heat form the source and convert it into equal amount of work done and no heat is given to the sink.
Mathematically,
Q1 ≠ W
Q2 ≠ O
2. Clausius Statement
According to Clausius Statement,
Without the performance of external work heat cannot flow from cold reservoir towards, the hot reservoir.
Example
In case of refrigerator flow of heat is unnatural but this unnatural flow of heat is possible only when we apply electrical power on the pump of the refrigerator.
Qs. Define the term Entropy and Give its Uses
ENTROPY
Definition
It measures the disorderness of any system.
Mathematically,
ΔS = ΔQ/T
Where Δs shows change in entropy.
Units
Joule per degree Kelvin – J/°K.
Explanation
As we know that incase of isometric process volume is constant. In case of Isothermal process temperature and pressure is constant, but in case of adiabatic process neither temperature, nor pressure or volume is constant but one thermal property is constant which is known as Entropy.
There are two types of Entropy.
1. Positive Entropy
2. Negative Entropy
1. Positive Entropy
If heat is supplied to the system the entropy will be positive.
2. Negative Entropy
When heat is rejected by the system the entropy will be negative.
Qs. What is carbot engine an carnot cycle?
CARNOT ENGINE
Definition
‘Carnot engine is an ideal heat engine which converts heat energy into mechanical energy.
Working of Carnot Engine
It consists of a cylinder and a piston. The walls of the cylinder are non-conducting while the bottom surface is the conducting one. The piston is also non-conducting and friction less. It works in four steps. Which are as follows.
1. Isothermal Expansion
2. Adiabatic Expansion
3. Isothermal Compression
4. Adiabatic Compression
1. Isothermal Expansion
First of all, cylinder is placed on a source and allow to move upward as a result temperature and pressure of the working substance decreases, while volume increases. In order to maintain temperature we have to supply more amount of heat from source to the cylinder. Since in this expansion temperature is kept constant.
2. Adiabatic Expansion
Secondly cylinder is placed on an insulator and piston is allow to move downward as a result temperature and pressure of working substance will decrease. While volume will increase but no heat is given or taken of the cylinder.
3. Isothermal Compression
In this state cylinder is placed on a sink and piston is allow to move downward as a result temperature and pressure of the working substance will increase while volume will decrease. In order to maintain temperature we have to reject extra heat from cylinder to the sink. Since in this compression temperature is constant.
4. Adiabatic Compression
Finally cylinder is placed on an insulator and piston is a flow to move downward, when we do so neither temperature nor pressure or volume is constant. But no heat is given or taken out of the cylinder.
CARNOT CYCLE
Definition
By combining the four processess Isothermal Expansion, Adiabatic Expansion, Isothermal Compression and Adiabatic Compression which are carried out in carnot engine, then we get a cycle knows as Carnot cycle.
Qs. How can we increase the efficiency of Heat Engine?
If we want to increase the efficiency of any heat engine then for this purpose we have to increase temperature of source as maximum as possible and reduce the temperature of sink as minimum as possible.
Qs. Define Specific Heat and Molar Specific Heat.
SPECIFIC HEAT
Definition
Specific heat is the amount of heat required to raise the temperature of a unit mass of a substance by one degree centigrade.
Different substances have different specific heat because number of molecules in one kg is different in different substances. It is denoted by c.
Mathematical Expression
Consider a substance having mass m at the temperature t1. The amount of heat supplied is ΔQ, which raises the temperature to t2. The change in temperature is Δt.
The quantity of heat is directly proportional to the mass of the substance.
ΔQ ∞ m
And the temperature difference
ΔQ ∞ Δt
Combining both the equations
ΔQ ∞ mΔt
=> ΔQ = cmΔt
=> c = ΔQ / mΔt —- (I)
Where c is the specific heat of the substance. Its unit is Joules / Kg°C.
MOLAR SPECIFIC HEAT
Definition
Molar specific heat is the amount of heat required to raise the temperature of one mole of a substance through one degree celsius.
Almost all the substances have the same amount of molar specific heat because the numbers of molecules in all substances are same in one mole. It is denoted by cM.
Mathematical Expression
Mathematically,
No. of Moles = Mass / Molecular Mass
=> n = m / M
=> nM = m
=> nM = ΔQ / nΔt
Where n is the number of moles. The unit of molar specific heat is J/Kg°C.
Qs. Define Molar Specific Heat at Constant volume and at Constant Pressure.
MOLAR SPECIFIC HEAT AT CONSTANT VOLUME
Definition
The amount of heat required to raise the temperature of one mole of any gas through one degree centigrade, at constant volume is known as molar specific heat volume.
It is denoted by Cv.
Mathematical Expression
Mathematically,
ΔQv = nCvΔt
Where ΔQv is the heat supplied at constant volume.
MOLAR SPECIFIC HEAT AT CONSTANT PRESSURE
Definition
The amount of heat required to raise the temperature of unit mass of a substance through one degree centigrade at constant pressure is known as Molar Specific Heat at Constant Pressure.
It is denoted by Cp.
Mathematical Expression
Mathematically,
ΔQp = nCpΔt
Where ΔQp is the heat supplied at constant volume.
ELECTROSTATICS
DEFINITION
Total Kinetic energy of a body is known as HEAT.
OR
Transfer of energy from a hot body to a cold one is termed as Heat.
Heat is measured by using an measurement centimeter.
UNITS
Since heat is a force of energy therefore its unit is Joule (J).
TEMPERATURE
DEFINITION
The average kinetic energy of a body is known as Temperature.
OR
The quantitative determination of degree of hotness may be termed as Temperature.
SCALES OF TEMPERATURE
There are three main scales of temperature.
1. Celsius Scale
2. Fahrenheit Scale
3. Kelvin Scale
Celsius and Fahrenheit scales are also known as Scales of Graduation.
1. Celsius Scale
The melting point of ice and boiling point of water at standard pressure (76cm of Hg) taken to be two fixed points. On the Celsius (centigrade) scale the interval between these two fixed points is divided into hundred equal parts. Each part thus represents one degree Celsius (1°C). This scale was suggested by Celsius in 1742.
Mathematically,
°C = K – 273
OR
°C = 5/9 (°F – 32)
2. Fahrenheit Scale
The melting point of ice and boiling of water at standard pressure (76cm of Hg) are taken to be two fixed points. On Fahrenheit scale the lower fixed point is marked 32 and upper fixed point 212. The interval between them is equally divided into 180 parts. Each part represents one degree Fahrenheit (1°F).
Mathematically,
°F = 9/5 (°C + 32)
3. Kelvin Scale
The lowest temperature on Kelvin Scale is -273°C. Thus 0° on Celsius scale will be 273 on Kelvin scale written as 273K and 100 on Celsius scale will be 373K. The size of Celsius and Kelvin scales are same.
Mathematically,
K = °C + 273
THERMAL EQUILIBRIUM
Heat flows from hot body to cold body till the temperature of the bodies becomes same, then they are said to be in Thermal Equilibrium.
THERMAL EXPANSION
DEFINITION
The phenomenon due to which solid experience a change in its length, volume or area on heating is known as Thermal Expansion.
Explanation
If we supply some amount of heat to any substance then size or shape of the substance will increase. This increment is known as Thermal Expansion. Thermal expansion is due to the increment of the amplitudes of the molecules.
Types of Thermal Expansion
There are three types of Thermal Expansion.
1. Linear Expansion
2. Superficial Expansion
3. Volumetric Expansion.
1. Linear Expansion.
If we supply some amount of heat to any rod, then the length of the rod, then the length of the rod will increase. Such increment is known as Linear Expansion.
2. Superficial Expansion.
If we apply some amount of heat to any square or rectangle then area of the square or rectangle will increase. Such increment is known as Superficial Expansion.
3. Volumetric Expansion.
If we apply some amount of heat to any cube, then the volume of the cube will increase. Such increment is known as Volumetric Expansion.
COEFFICIENT OF LINEAR EXPANSION
CONSIDERATION
Let Lo be the initial length of rod at t1 °C. If we increase the temperature from t1 °C to t2 °C, then length of the rod will increase. This increment in length is denoted by ΔL. The increment in length depends upon the following two factors.
1. Original Length (Lo)
2. Difference in temperature Δt
Derivation
The increment in length is directly proportional to the original length and temperature difference.
Mathematically,
ΔL ∞ Lo —– (I)
ΔL ∞ Δt —– (II)
Combining eq (I) and (II), we get
ΔL ∞ LoΔt
=> ΔL = ∞LoΔt
Where α is the constant of proportionality and it is known as coefficient of Linear Expansion. It is defined as,
It is the increment in length per unit length per degree rise in temperature.
Its unit is 1/°C or °C. If Lt is the total length, then
Lt = Lo + ΔL
=> Lt = Lo + αLoΔt
=> Lt = Lo (1 + αΔt)
COEFFICIENT OF VOLUMETRIC EXPANSION
Consideration
Let Vo be the initial length of rod at t1 °C. If we increase the temperature from t1°C to t2°C then length of the rod will increase. This increment in length is denoted by ΔV. The increment in length depends upon the following two factors.
3. Original Volume (Lo)
4. Difference in temperature Δt
Derivation
The increment in volume is directly proportional to the original volume of temperature difference.
Mathematically,
ΔV ∞ Vo —- (I)
ΔV ∞ Δt —- (II)
Combining eq (I) and (II), we get,
ΔV ∞ Vo Δt
=> ΔV = βVoΔt
Where β is the constant of proportionality and it is known as coefficient of Volumetric Expansion. It is defined as
It is the increment in volume per unit volume per degree rise in temperature.
Its unit is 1/°C or °C-1. If Vt is the total volume then
Vt = Vo + ΔV
=> Vt = Vo + αβVo Δt
=> Vt = Vo (1 + βΔt)
State and Explain Boyle’s Law and Charle’s Law.
INTRODUCTION
Gas Laws are the laws, which give relationship between Pressure, Volume, temperature and mass of the gas. There are two gas laws.
1. Boyle’s Law
2. Charle’s Law
BOYLE’S LAW
Statement 1
According to first statement of Boyle’s Law:
Volume of the known mass of gas is inversely proportional to the pressure, if temperature is kept constant.
Mathematical Form
Mathematically,
V ∞ 1/P
=> V = K 1/P
=> PV = K (Constant)
P1V1 = P2V2 = … = K
=> P1V1 = P2V2
The above equation is mathematical form of Boyle’s Law.
Statement II
According to second statement of Boyle’s Law.
The product of the pressure and volume of the known mass of the gas remain constant if the temperature is kept constant.
Statement III
According to third statement of Boyle’s Law.
The product of pressure and volume of a gas is directly proportional to the mass of a gas, provided that temperature is kept constant.
Mathematical Form
Mathematically,
PV ∞ m
=> PV = Km
=> PV/m = K
=> P1V1/m1 = P2V2/m2
Limitations of Boyle’s Law
Boyle’s Law does not hold good at high pressure, because at high pressure gases convert into liquid or solid.
Graphical Representation
The graph between pressure and volume is a curved line, which shows that volume and pressure are inversely proportional to each other.
CHARLE’S LAW
Statement I
According to first statement of Charle’s Law.
Volume of known mass of gas is directly proportional to the absolute temperature, if then pressure is kept constant.
Mathematical Form
Mathematically,
V ∞ T
=> V = KT
=> V/T = K
OR
=> V1/T1 = V2/T2
The above equation is mathematical form of Charles Law.
Statement II
According to second statement of Charles Law.
The ratio between volume and temperature of the known mass of a gas is always constant, if pressure is kept constant.
Limitations of the Law
This law does not hold good at low temperature because at low temperature gases convert into liquid or solid.
GENERAL GAS EQUATION
It is the combination of Boyle’s law, Charle’s Law and Avogadro’s Law. According to Boyle’s Law.
V ∞ 1/P —- (I)
According to Charle’s Law
V ∞ T —- (II)
According to Avogadro’s Law
V ∞ n —- (III)
Combining eq (I), eq (II) and eq (III)
V α nT/P
=> V = RnT/P
=> PV = RnT —- (A)
Where R is the universal gas constant, We Know that
R = R/NA
=> R = KNA
Where K is the Boltzman constant, Its value is
K = 1.38 x 10(-23) J/K
Substituting the value of R in eq (A)
=> PV = nKNAT
=> PV = nNAKT
But nNA = N1 (Total number of molecules), therefore,
PV = NtKT
=> P = Nt/V KT
Since Nt/V = N (Total Number of molecules in a given volume), therefore,
P = NKT
The above equation is other form of General Gas Equation.
Qs. What are the basic postulates of Kinetic Molecular Theory pf Gases?
INTRODUCTION
The properties of matter in bulk can however be predicted on molecular basis by a theory known as Kinetic Molecular theory of gases. The characteristic of this theory are described by some fundamental assumptions, which explained below:
BASIC POSTULATES OF KINETIC MOLECULAR THEORY OF GASES
1. Composition
All gases are composed of small, spherical solid particle called molecules.
2. Dimension of Molecules
The dimensions of the molecules is compared to the separation between the molecule is very small.
3. Number of Molecules
At standard condition, there are 3 x 10(23) molecules in a cubic meter.
4. Pressure of Gas
Gas molecules collide with each other as well as with the wall of the container and exert force on the walls of the container. This force per unit are is known as Pressure.
5. Collision Between the Molecules
The collision between the molecules is elastic in which momentum and Kinetic energy remains constant.
7. Kinetic Energy of Molecules
If we increase the temperature of gas molecules, then K.E will also increase. It means that average kinetic energy of the gas molecules is directly proportional to the absolute temperature.
8. Forces Of Interraction
There is no force of attraction or repulsion between the molecules.
9. Law of Mechanics
Newtonian mechanics is applicable to the motion of molecules.
THERMODYNAMICS
DEFINITIONS
The branch of Physics that deals with the conversion of heat energy into mechanical energy or work or transformation of work into heat energy is known as Thermodynamics.
Laws of Thermodynamics
There are two laws of thermodynamics.
1. First Law of Thermodynamics
2. Second Law of Thermodynamics
State and explain first law of Thermodynamics. What are the application of first law of Thermodynamics?
FIRST LAW OF THERMODYNAMICS
First Statement
Whenever heat energy is converted into work or work is transformed into heat energy, the total amount of heat energy is directly proportional to the total amount of work done.
Mathematical Expression
Mathematically,
Q ∞ W
=> Q = JW
Where J is the mechanical equivalent of heat or joules constant. Its value is 4.2 joules.
Second Statement
If ΔQ is the amount of heat supplied to any system, then this heat will be utilized to increase the internal energy of the system in the work done in order to move the piston.
Mathematical Expression
Mathematically,
ΔQ – Au + Δw
The above equation is the mathematical form of first law of thermodynamics.
Where
Δu = Internal energy of the system.
Δw = Amount of work done.
ΔQ will be positive when heat is supplied to the system and it is negative when heat is rejected by the system.
Δw will be positive when work is done by the system and it will be negative when work is done on the system.
Third Statement
For a cyclic process, the heat energy supplied to a system and work done on the system is equal to the sum of heat energy rejected by the system.
Mathematical Expression
Mathematically,
Q(IN) + W(IN) = Q(OUT) + W(OUT)
Q(IN) – Q(OUT) = W(OUT) + W(IN)
ΔQ = ΔW
{dQ = {dW
{Shows cyclic process
Fourth Statement
For a system and surrounding the total amount of heat energy remains constant
APPLICATIONS OF THE LAW
There are four applications of first law of Thermodynamics.
1. Isometric or Isocohric Process.
2. Isobaric Process
3. Isothermal Process
4. Adiabatic Process
1. Isometric or Isocohric Process
The process in which volume of the system remains constant is known as Isometric Process.
In this process all supplied amount of heat is utilized to increase the internal energy of the system.
Mathematical Form
In this process first law of thermodynamics take the following form.
ΔQ = Δu + ΔW
But,
ΔW = 0
=> ΔQ = Δu = 0
=> ΔQ = Δu
2. Isobaric Process
The process in which pressure is kept constant is known as isobaric process.
In this process, all supplied amount of heat is utilized for the following two functions.
i. To increase the internal energy of the system.
ii. In work done in order to move the piston upward.
3. Isothermal Process
A process in which temperature is kept constant is known as Isothermal Process.
There are two parts of isothermal process.
i. Isothermal Expansion
ii. Isothermal Compression
i. Isothermal Expansion
In this process cyclinder is placed on a source and piston is allowed to move upward. When we do so temperature and pressure of the working substance will decrease while volume will increase. In order to keep the temperature constant, we have to supply required amount of heat from source to cylinder.
Since in this expansion, temperature is constant therefore it is known as Isothermal Expansion.
ii. Isothermal Compression
In this process, cylinder is placed on a sink and piston is allowed to move downward. When we do so temperature and pressure of working substance will increase while volume will decrease. In order to maintain the temperature, we have to reject required amount of heat from cylinder to the sink.
Since in this compression, temperature is kept constant therefore it is known as isothermal compression.
SECOND LAW OF THERMODYNAMICS
Introduction
It is inherit tendency of heat that it always flows from hot reservoir to cold reservoir. Rather than to flow in both the directions with equal probability. On the basis of this tendency of heat a law was proposed that is known as Second Law of Thermodynamics.
Statement
It is impossible to construct a process which reserves the natural tendency of heat.
This law is also known as Law of heat and can also be stated as
Efficiency of heat engine is always less than unity.
Explanation
Many statements of this law has been proposed to cover similar but different point of vies in which two are given below.
1. Lord Kelvin Statement
2. Clausius Statement
1. Lord Kelvin Statement
According to this statement,
It is impossible to construct a heat engine which extract all heat form the source and convert it into equal amount of work done and no heat is given to the sink.
Mathematically,
Q1 ≠ W
Q2 ≠ O
2. Clausius Statement
According to Clausius Statement,
Without the performance of external work heat cannot flow from cold reservoir towards, the hot reservoir.
Example
In case of refrigerator flow of heat is unnatural but this unnatural flow of heat is possible only when we apply electrical power on the pump of the refrigerator.
Qs. Define the term Entropy and Give its Uses
ENTROPY
Definition
It measures the disorderness of any system.
Mathematically,
ΔS = ΔQ/T
Where Δs shows change in entropy.
Units
Joule per degree Kelvin – J/°K.
Explanation
As we know that incase of isometric process volume is constant. In case of Isothermal process temperature and pressure is constant, but in case of adiabatic process neither temperature, nor pressure or volume is constant but one thermal property is constant which is known as Entropy.
There are two types of Entropy.
1. Positive Entropy
2. Negative Entropy
1. Positive Entropy
If heat is supplied to the system the entropy will be positive.
2. Negative Entropy
When heat is rejected by the system the entropy will be negative.
Qs. What is carbot engine an carnot cycle?
CARNOT ENGINE
Definition
‘Carnot engine is an ideal heat engine which converts heat energy into mechanical energy.
Working of Carnot Engine
It consists of a cylinder and a piston. The walls of the cylinder are non-conducting while the bottom surface is the conducting one. The piston is also non-conducting and friction less. It works in four steps. Which are as follows.
1. Isothermal Expansion
2. Adiabatic Expansion
3. Isothermal Compression
4. Adiabatic Compression
1. Isothermal Expansion
First of all, cylinder is placed on a source and allow to move upward as a result temperature and pressure of the working substance decreases, while volume increases. In order to maintain temperature we have to supply more amount of heat from source to the cylinder. Since in this expansion temperature is kept constant.
2. Adiabatic Expansion
Secondly cylinder is placed on an insulator and piston is allow to move downward as a result temperature and pressure of working substance will decrease. While volume will increase but no heat is given or taken of the cylinder.
3. Isothermal Compression
In this state cylinder is placed on a sink and piston is allow to move downward as a result temperature and pressure of the working substance will increase while volume will decrease. In order to maintain temperature we have to reject extra heat from cylinder to the sink. Since in this compression temperature is constant.
4. Adiabatic Compression
Finally cylinder is placed on an insulator and piston is a flow to move downward, when we do so neither temperature nor pressure or volume is constant. But no heat is given or taken out of the cylinder.
CARNOT CYCLE
Definition
By combining the four processess Isothermal Expansion, Adiabatic Expansion, Isothermal Compression and Adiabatic Compression which are carried out in carnot engine, then we get a cycle knows as Carnot cycle.
Qs. How can we increase the efficiency of Heat Engine?
If we want to increase the efficiency of any heat engine then for this purpose we have to increase temperature of source as maximum as possible and reduce the temperature of sink as minimum as possible.
Qs. Define Specific Heat and Molar Specific Heat.
SPECIFIC HEAT
Definition
Specific heat is the amount of heat required to raise the temperature of a unit mass of a substance by one degree centigrade.
Different substances have different specific heat because number of molecules in one kg is different in different substances. It is denoted by c.
Mathematical Expression
Consider a substance having mass m at the temperature t1. The amount of heat supplied is ΔQ, which raises the temperature to t2. The change in temperature is Δt.
The quantity of heat is directly proportional to the mass of the substance.
ΔQ ∞ m
And the temperature difference
ΔQ ∞ Δt
Combining both the equations
ΔQ ∞ mΔt
=> ΔQ = cmΔt
=> c = ΔQ / mΔt —- (I)
Where c is the specific heat of the substance. Its unit is Joules / Kg°C.
MOLAR SPECIFIC HEAT
Definition
Molar specific heat is the amount of heat required to raise the temperature of one mole of a substance through one degree celsius.
Almost all the substances have the same amount of molar specific heat because the numbers of molecules in all substances are same in one mole. It is denoted by cM.
Mathematical Expression
Mathematically,
No. of Moles = Mass / Molecular Mass
=> n = m / M
=> nM = m
=> nM = ΔQ / nΔt
Where n is the number of moles. The unit of molar specific heat is J/Kg°C.
Qs. Define Molar Specific Heat at Constant volume and at Constant Pressure.
MOLAR SPECIFIC HEAT AT CONSTANT VOLUME
Definition
The amount of heat required to raise the temperature of one mole of any gas through one degree centigrade, at constant volume is known as molar specific heat volume.
It is denoted by Cv.
Mathematical Expression
Mathematically,
ΔQv = nCvΔt
Where ΔQv is the heat supplied at constant volume.
MOLAR SPECIFIC HEAT AT CONSTANT PRESSURE
Definition
The amount of heat required to raise the temperature of unit mass of a substance through one degree centigrade at constant pressure is known as Molar Specific Heat at Constant Pressure.
It is denoted by Cp.
Mathematical Expression
Mathematically,
ΔQp = nCpΔt
Where ΔQp is the heat supplied at constant volume.
ELECTROSTATICS
CHAPTER – 12
Qs. State and Explain Coulomb’s Law.
INTRODUCTION
In 1974, Sir Augusts de coulomb studied the relationship between localized charges. He carried out experiments using torsion balance. In the basis of his experimental results, he proposed a law known as Coulomb’s Law.
Statement
The electric force of interaction between two point charges is directly proportional to product of their charges and inversely proportional to the square of the distance from their centers.
Mathematical Expression
Consider two points charges q1 and q2 and let “1″ be the separation between them. According to Coulomb’s Law.
F ∞ q1 q2 ——– (i)
And, according to second part
F ∞ 1/r² ——— (ii)
Combining eq (i) and eq (ii), we get
F ∞ q1 q2 / r²
F = K q1 q2 / r²
Where K is the constant of proportionality and its value depends upon the medium between the two charges. In S.I system its value is equal to
K = 8.98 x 10(9) N-m² / c².
Coulomb’s Law can be expressed in terms of permitivity as follows.
F = 1 / 4 π Є . q1 q2 / r²
Where Єo is the permitivity of free space and its value is
Єo = 8.85 x 10(-12) col² / Nm²
if some other medium is used instead of air then
F = 1 / 4 π Єo Єr . q1 q2 / r²
Where Єr is the relative permitivity which is different for different dielectric.
Qs. Define electric field and electric intensity. Find the formula for the electric intensity due to a point charge.
ELECTRIC INTENSITY
DEFINITION
Electric intensity is the force experienced by a unit positive charge due to the presence of a charge body.
EXPLANATION
It is the measure of the strength of the electric field. Electric intensity is a vector quantity and its direction is same as that of force. If the charge is positive then electric intensity is directed from the charge and if the charge is negative, then it is directed towards the charge. The SI unit of electric intensity is Newton per Coulomb or volt per meter.
Mathematical Form
Mathematically electric intensity is given by
E = F / q°
Where E = Electric Intensity
F = Force
q° = small magnitude
ELECTRIC INTENSITY DUE TO A POINT CHARGE
CONSIDERATION
Consider a charged body q that has an electric field all around it. We want to find out electric intensity E at point P. For this purpose, we placed a point charge q1 at that point.
Qs. Define Electric flux. Find the formula for the electric flux due to point charge in a closed sphere.
ELECTRIC FLUX
DEFINITION
The total number of imaginary lines drawn in such a way that the attraction of tangent at any point is same as that of electric field crossing a surface normally is called electric flux or flux on the surface.
OR
The total number of lines of forces crossing a surface normally is called flux on that surface.
Mathematical Form
The flux at a surface is determined by the product of flux density i.e. electric field and the projection of its area perpendicular to the field or by the product of area and component of field normal to the area.
ELECTRIC FLUX DUE TO A POINT CHARGE IN A CLOSED SPHERE
Consider an isolated point charge +q. Th lines of forces from q will spread uniformly in space around it cutting the surface of an imaginary sphere. Now we want to find flux due to point charge. For this purpose, we divide the whole sphere into small patches. Each patch is denoted by ΔA.
Qs. State and prove Guass’s Law
The total electric flux diverging out from a closed surface is equal to the product of the sum of all charges present in that closed surface and 1/ Єo.
Find the Formula for Electric Intensity.
1. Due to Charge Sheet
2. Due to Two Charge Sheets
OR
What are the application of Guass’s Law.
Introduction
Guass’s Law can be used to calculate the electric field only in those cases of charge distribution which are so symmetrical that by proper choice of Guassian surface the flux on it may possibly be evaluated. With the help of guass’s law we can measure the intensity in following cases.
1. Electric Intensity due to charge sheet.
2. Electric Intensity due to two charge sheets.
1. Electric Intensity Due to Charge Sheet.
Consider a charge sheet in which unit positive charges are uniformly distributed.
As we know that charge density is the charge stored per unit area and is denoted by
σ = Q / A
=> Q = σ A
According to Guass’s Law
σ = Q / Є
=> φe = σ A / Є ——– (I)
Now consider a cylindrical shell, which is placed inside the charge sheet. It has three surfaces.
Upper Surface
Curve Surface
Bottom
Electric Intensity Due to Upper Surface
φ1 = E ΔA
φ1 = E ΔA cos θ
But θ = 0° and cos 0° = 1, therefore,
φ1 = E ΔA cos 0°
= φ1 = E ΔA
Electric Intensity Due to Curved Surface
φ2 = E ΔA
φ2 = E ΔA cos θ
Since the angle between the field vector and area vector of all elements of curved surface is 90°, therefore,
φ2 = E ΔA cos 90°
But cos 90° = 0,
φ2 = E ΔA (0)
φ2 = 0
Electric Intensity Due to Lower Surface
φ3 = E ΔA
φ3 = E ΔA cos θ
But 0 = 0° and cos 0°, = 1 therefore,
φ3 = E ΔA cos 0°
=> φ3 = E ΔA
Total Flux
Total Flux is given by
φe = φ1 + φ2 + φ3
=> φe = E ΔA + 0 + E ΔA
=> φe = 2E ΔA
For the whole charge sheet:
=> φe = 2E Σ ΔA
=> φe = 2 EA ———– (II)
Comparing equation (I) and (II)
=> σ A / Єo = 2AE
=> E = σ / 2 Єo
OR
E = σ / 2 Єo
2. Electric Intensity Due to Two Charge Sheets.
As we know that electric intensity due to a charge sheet is
E = σ / 2 Єo
For two sheets E will be,
=> E = σ / 2 Єo + σ / 2 Єo
=> E = σ + σ / 2 Єo
=> E = 2 σ / 2 Єo
=> E = σ / Єo
=> E = σ / Єo i
Qs. Define the term Capacitor and Capacitance of a Capacitor
CAPACITOR
Capacitor is a device which is use to storage charge. A simple capacitor consists of two parallel metallic plates. A plate is connected to the positive terminal of the battery and another plate is connected to the negative terminal of the battery.
CAPACITANCE
Definition
The capacity of a capacitor to store the charge is known as Capacitance.
Mathematical Explanation
If V is the voltage provided to the capacitor and Q is the amount of charge stored in the capacitor, then it is observed that if more is the voltage, then more will be the charge stored in the capacitor.
Mathematically,
Q ∞ V
=> Q = CV
Where C is the capacitance of the capacitor which may be defined as
The ratio of the charge on one of the plate (conductor) to the potential difference between them.
Unit of Capacitance
The unit of Capacitance is Farad. It may be defined as,
If one coulomb charge is stored due to 1 volt potential, then capacitance will be 1 Farad.
Factors on which Capacitance Depends.
The Capacitance of a capacitor depends upon following factors.
1. Cross-sectional area of plate
2. Separation between plates
3. Dielectric
On the basis of dielectric capacitors are classified into different types. For example Electrolyte Capacitor, Paper Capacitor, Meca Capacitor, Oil Capacitor.
Types of Capacitor
There are two main types of Capacitors.
1. Fixed Capacitor
2. Variable Capacitor
1. Fixed Capacitor
Those capacitor whose capacitance is constant are known as Fixed Capacitor. For example Paper Capacitor and Meca Capacitor.
2. Variable Capacitor
Those capacitor whose capacitance is not fixed are known as Variable Capacitor. For example Gang Capacitor.
Find the Formula for the capacitance of Parallel plate capacitor.
PARALLEL PLATE CAPACITOR
Definition
A parallel plate capacitor is a device used to store the charge. It consist of two parallel metallic plates. A plate is connected to the positive terminal of the battery and another plate is connected to the negative terminal of the battery.
These plates are separated by a very small distance compared to the dimension of the plates.
CAPACITANCE OF A PARALLEL PLATE CAPACITOR
Consideration
Consider a parallel plate capacitor in which is the distance between the plates the charge stored is denoted by Q where as potential difference between the plates is V.
Derivation of the Formula
As we known that electric intensity due to two charge sheets is
E = σ / Єo
Since charge density is given by
σ = Q / A
Substituting the value in above equation
E = Q / A Єo ——— (I)
According to the definition of electrical potential.
ΔV = E Δ
Substituting the value of E from eq (I) in above equation
Q = C x Qd / A Єo
C = A Єo / d
If some other medium is used instead of air then,
C = A Єo Єr / d
Find the formula for equivalent capacitance when,
1. Capacitors are connected in Series
2. Capacitors are connected in Parallel
Introduction
Capacitors of some fixed values are used in a circuit. The capacitance of the desired value can however be obtained by suitable combination of capacitor. Capacitors can be combined in parallel, series or both.
WHEN CAPACITORS ARE CONNECTED IN SERIES OR SERIES COMBINATION
Consideration
Consider three capacitors having capacitance C1, C2, C3 connected in a series. These capacitors can be replace by an equivalent capacitor having capacitance Ce. When a cell is connected across the ends of system then a charge Q is transferred across the plates of capacitors. A charge upon one plate always attracts upon the other plate with a charge equal in magnitude and opposite in sign.
Let V be the potential difference across the combination. The potential difference across the individual capacitor is Vab, Vbc, Vcd.
Derivation of the Formula
As we know that in case of series combination the charge across the individual capacitor remains constant, where as potential difference varies such that the potential difference V is the sum of potential difference applied across individual capacitor.
Vad = Vab + Vbc + Vcd ———— (I)
We know that,
Q = CV
V = Q / C
Therfore,
Vad = Q / Ce
Vab = Q / C1
Vbc = Q / C2
Vcd = Q / C3
Substituting the values of Vad, Vab, Vbc and Vcd in eq (I)
Q/Ce = Q/C1 + Q/C2 + Q/C3
Q/Ce = Q [1/C1 + 1/C2 + 1/C3]
1/Ce = 1/C1 + 1/C2 + 1/C3
Conclusion
The reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitance.
WHEN CAPACITORS ARE CONNECTED IN PARALLEL OR PARALLEL COMBINATION
Consideration
Consider three capacitors having C1, C2, C3 capacitance respectively are connected in parallel. We can replace them by an equivalent capacitor having capacitance Ce. A charge q given to a point divide it self reside on the plates of individual capacitor as Q1, Q2, Q3 respectively.
Derivation of the Formula
As we known that in case of parallel combination the potential difference across each capacitor is that of the source where as the charge across each capacitor varies, therefore the total charge Q is given by
Q = Q1 + Q2 + Q3 ——— (I)
As we known that,
Q = CV
Therefore,
Q = Ce Vab
=> Q1 = C1 Vab
=> Q2 = C2 Vab
=> Q3 = C3 Vab
Substituting the values of Q1, Q2, Q3 in eq (I)
Ce Vab = C1 Vab + C2 Vab + C3 Vab
Ce Vab + Vab (C1 + C2 + C3)
Ce = C1 + C2 + C3
Conclusion
The equivalent capacitance is equal to sum of the individual capacitances.
Qs. Define electric potential and absolute potential.
ELECTRIC POTENTIAL
Definition
Electric Potential is the amount of work done in order to bring a unit positive charge from one point to another point against the direction of electric field.
Explanation
In order to bring a unit positive charge from one point to another point, we have to do some work. This work is stored in the form of potential energy. This potential energy per unit charge is known as Electric Potential.
Mathematical Form
Electric Potential is denoted by V. It is defined as the potential energy per unit charge. Therefore mathematically,
ΔV = Δu / q ——- (I)
We know that
Δu = F. Δr
But E = F/q => F = Eq, therefore,
Δu = EqΔr
Substituting the value in eq (I)
ΔV = EqΔr / q
=> ΔV = E. Δr
Where,
ΔV = Electric Potential
E = Electric Intensity
Δr – Displacement
From above expression we know that
Electric Potential is the dot product of electric intensity and displacement.
It is a scalar quantity and its unit is Volt.
ABSOLUTE POTENTIAL
Definition
Absolute potential is the amount of work done in order to bring a unit positive charge from one point to infinite distance against the direction of magnetic field.
ELECTRIC POTENTIAL
Electric Potential is the amount of work done in order to move a unit positive charge from one point to another against the direction of electric.
Consideration
Consider a unit positive charge placed in a uniform electric field. We have to displace it from point O to point N. For this purpose we have to do some work. This work is known as electric potential. In order to determine electric potential from point O to point N we divide the whole distance into small equal patches because in a long distance intensity does not remain constant. This patch is denoted by Δr.
CURRENT ELECTRICITY
CHAPTER – 13
Qs. Define Charge and Current.
CHARGE
Definition
Flow of electron is known as Charge.
It is denoted by Q.
Unit
Its unit is Coulomb.
1 Coulomb = 10(-6) μcoulomb
1 coulomb = 10 (-3) mili coulomb
1 coulomb = 10(-9) neno coulomb
CURRENT
Definition
The flow of charge per unit time is known as Current.
It is denoted by I.
Unit
The unit of current is coulomb/sec or Ampere.
AMPERE
If one coulomb charge passes through the conductor in 1 second then the current is 1 Ampere.
Mathematical Form
Mathematically,
I = Q/t
Qs. State and Explain Ohm’s Law.
OHM’S LAW
Introduction
A German scientist George Simon Ohm studied the relationship between voltage, current and resistance. On the basis of his experimental results, he proposed a law which is known as Ohm’s Law.
Statement
Ohm’s Law to metallic conductors can be stated as
The current through a conductor is directly proportional to the potential difference between the ends of the conductor provided that physical conditions are kept constant.
It can also be stated as
The ratio between voltage and current remains constant, if the physical conditions are kept constant.
Mathematical Form
Mathematically,
V ∞ I
V = IR
R = V/I
Where R is the constant of proportionality known as resistance of the conductor. Its unit is volt per ampere (Volt/Ampere) or Ohm (Ω).
Ohm (Ω)
If 1 ampere current passes through the conductor due to 1 volt potential difference then the resistance of conductor is 1 Ohm.
Resistance
Opposition offered in the flow of current.
Graphical Representation.
When graph is plotted between current and potential differences then straight line is obtained.
Limitations of the Law
Ohm’s Law is valid only for metallic resistance at a given temperature and for steady currents.
Qs. Define the term Resistivity or Coefficient of Resistor.
RESISTIVITY OR COEFFICIENT OF RESISTOR
Definition
It is the resistance of a unit conductor whose cross-sectional area is 1 sqm.
Unit
Its unit is Ohm meter.
Mathematical Form
The resistance of any conductor depends upon the following factors.
1. Length of the conductor
2. Cross-sectional area of the conductor.
3. Material of the conductor.
The resistance of the conductor is directly proportional to the length of the conductor and inversely proportional to the cross-sectional area.
Mathematically,
R ∞ L ——– (I)
R α 1/A —— (II)
Combining eq (I) and (II)
R α L/A
=> R = ρL/A
Where ρ is the constant of proportionality known as Resistivity or Coefficient of resistance.
ρ = RA/L
Qs. Explain the effect of temperature on resistance or temperature coefficient of resistance.
EFFECT OF TEMPERATURE ON RESISTANCE
It is observed that if we increase the temperature then resistance of a conductor will increase.
Consideration
Let Ro be the initial resistance of a conductor at 4°C. If we increase the temperature from t1°C to t2°C, then resistance will increase. This increment in resistance is denoted by ΔR. The increment in resistance depends upon the following two factors.
1. Original Resistance (Ro)
2. Difference in temperature Δt.
Mathematical Verification
The increment in resistance is directly proportional to the original resistance and temperature difference.
Mathematically,
ΔR ∞ Ro —– (I)
ΔR ∞ Δt —– (II)
Combining eq (I) and eq (II) we get
ΔR ∞ RoΔt
=> ΔR = αRoΔt
Where α is the temperature coefficient of resistance. It is defined as
It is the increment in resistance per unit resistance per degree rise in temperature.
Its unit is 1/°C or °C. If RT is the total resistance, then
RT = Ro + ΔR
=> RT = Ro + αRo Δt
=> RT = Ro (1 + αΔt)
As we know that resistance is directly proportional to resistivity therefore,
ρT = ρo (1 + αΔt)
Qs. Define the term Power Decipation in Resistor.
POWER DECIPATION IN RESISTORS
Definition
When current flows in a conductor then a part of electrical energy appears in the form of heat energy which is known as Power Decipation in Resistor.
Units
Its unit is Joule per second (J/s). Most commonly used unit is Kwh.
1 Kwh = 36 x 10(5) Joules
Mathematical Form
Since,
P = Electrical Work / Time
Electrical Work = QV —— (I)
This electrical work produces heat energy in the resistor.
P = QV / t
P = Q / t . V
But,
I = Q / t
P = VI
From Ohm’s Law
V = IR
P = IIR
P = I2R
OR,
P = 12R2 / R
=> P = V2 / R
As we know that,
Energy = Power x time
=> E = P x t
=> E = Vit
=> E = I2Rt
And,
E = V2 / R . t
Qs. Define and explain Electromotive Force.
ELECTROMOTIVE FORCE
Definition
It is the terminal voltage difference when no current draws from a cell or a battery.
OR
Work done per coulomb on the charges.
It is denoted by E.
Unit
Electromotive force or simply e.m.f is a scalar quantity it has the same dimension as that of voltage, therefore its unit is volt.
Explanation
When an electric current passes through a resistor, it dissipates energy, which is transformed into heat energy. Thus to sustain a current in conductor some source of energy is needed so that it could continuously supply power equal to that which is dissipated as heat in the resistor. The strength of this source is called Electromotive Force.
Consideration
Let consider a simple circuit in which a resistor “R” is connected by leads of negligible resistance to the terminals of a battery. The battery is made up of some electrolyte and electrode for the production of e.m.f and hence when this current flows from battery, it encounters some resistance by the electrolyte present in two electrodes. This resistance is known as internal resistance “r” of the battery.
Mathematical Form
According to Ohm’s Law
V = IR
I = V / R
Or,
I = E / R + r
Where E is e.m.f and r is internal resistance
=> E = IR + Ir
E = V + Ir
MAGNETISM AND ELECTRO-MAGNETISM
CHAPTER-14
MAGNETIC FIELD DUE TO CURRENT
It was discovered by Oersted that when current masses through a conductor, magnetic field is produced. This field is known as “Magnetic Field of Induction” and is denoted bu “B”.
Ampere found that when two current carrying conductors are near each other, they experience force at each other. If the current is in the same direction the force is attractive and if the current is in opposite direction.
When electric charges are at rest they exert electrostatic force of attraction or repulsion on each other. When the charges are in motion they exert electric as well as magnetic force on each other because and isolated moving positive and negative charge create both electric and magnetic field.
MAGNETIC FIELD
Magnetic Field is a space or region around a magnet or current carrying coil of wire where its effect can be felt by small compass needle. Magnetic field of induction can be visualized by magnetic lines of induction.
A line of induction is an endless curve, which can be traced by a compass needle.
MAGNETIC FLUX AND FLUX DENSITY
The total number of magnetic lines of induction passing through a surface is called magnetic flux.
DETERMINING THE CHARGE TO MASS RATIO OF AN ELECTRON
The charge to mass into of an electron was determined by Sir J.J. Thomson by an apparatus which consists of a highly evacuated pear shaped glass pulls into which several metallic electrodes are sealed.
Electrons are produced by heating a tungsten flament F by passing a current through it. The electrons moving sideways are also directed towards the screen by applying negative potential on a hollow cylinder C open on both the sides surrounding the filament. Electrons are accelerated by applying positive potential to discs A and B. If V be the total total P.d between the disc Band the filament F taken then Kinetic Energy.
The beam strikes the screen coated with zinc sulphide after passing through the middle of the two horizontal moetal P’P and a spot of the light produced at O on the screen where the beam strikes and its position is noted.
A magnetic field of induction B is produced in between the plate directed into the paper. The magnetic field is produced by two identical current carrying coils placed on either side of the tube at the position of plates.
The force due to the magnetic field on the moving electron makes them move in a curved path and the light spot shifts from O to O on the screen there from of magnetic field acts as centripetal force
e. V B = mv2 / r
e/m = V/Br ——– I
e/m can be computed if the radius r and the expression of the circular path are in which the beam moves in the field region is determined. The radius r is calculated from the shift of the light spot i.e. r = 3.
A better method of determined V is as under. An electric field E is produced between the plates by applying suitable potential difference to exert a force “Be” on the electron opposite to that due to the magnetic field.
The potential diff. VI is so adjusted that two fields neutralize each other effects and the spot come back to its initial position O. Thus each other effects and the spot come back to its initial position O. Thus
Ee = Be V
Or
V = E/B —– (II)
Where E = V1 / d
d = distance between the plates.
Putting the value of V from eq 2 in 1
e/m = E/B2r
e/m = K75888 x 10(11) e/kg
AMPERE’S LAW
According to this law the sum of the product of the tangential component of the magnetic field of indaction and te length of an element of a closed curve taken in a magnetic field is μo times the current which passes through that area bounded by the curve.
Consider a long straight wire carrying a current 1 in the direction. The lines of force are concentric circles with their common centre on the wire. From these circles consider a circle of radius r. The magnitude of the magnetic field at all points on this circle and inside the circle is same.
Biot and Savart experimentally found that the magnitude of the field depends directly on twice the current and inversely proportional to the distance r from the conductor.
SOLENOIDAL FIELD
A solenoid is a coil of an insulated copper wire wound on a circular cylinder with closed turns. When current passes through it, magnetic field is produced with is uniform and strong inside the solenoid while outside it the field is negligibly weak.
Consider a solenoid through which the current 1 is passing in order to determine the magnetic field of induction B at any point inside the solenoid imagine a closed path “abcda” on the form of a rectangular. The rectangular is divided into four elements of length L1, L2, L3, L4. L1 is along the axis inside the solenoid and L3 is far from the solenoid.
By applying amperes circuital law
B L1 + B. L2 + L2 + B. L3 + B. L4 = μo x current enclosed —– (I)
Since B. L1 is parallel inside the solenoid
B. L1 = BL4 cos 0 = BL4
The field is very weak outside the solenoid is very weak and therefore it can be negnected thus
B. L3 = 0
As B is perpendicular to L2 and L4 inside the solenoid therefore
B. L2 = BL2 cos 90 = 0
B. L4 = BL4 cos 90 = 0
substitute the above values is eq 1
B. L1 + O + 0 + 0 = μo x current closed
B. L1 = μo x current enclosed ——- (II)
If there are n turns per unit length of the solenoid and each turn carries a current I will be “n L1I”
TOROIDAL FIELD
A Toroid or a circular solenoid is a coil of insulated copper wire wound on a circular core with close turn. When the current passes through the toroid, magnetic field is produced which is strong enough inside while outside it is almost zero.
Consider a toroid that consists of N closely packed turns that carry a current I. Imagine a circular curve of concentric the core.
It is evident form of the symmetry at all points of the curve must have the same magnitude an should be tangential to the curve at all points. Divide the circle into small elements each of length ΔL is so small that B and ΔL are parallel to each other.
By amperes law
Σ B : ΔL = μo x current enclosed
ΣB ΔL Cos 0 = μo x current enclosed
ΣB ΔL = μo x current enclosed
BΣ ΔL = μo x current enclosed
Σ ΔL = 2 π r
B 2 π r = μo x current enclosed ——– (I)
Cases
If the circular path 1 is outside the core on the inner side of the toroid if enclose no current. Thus eq 1 become
B 2 π r = μo x 0 = 0
B = 0
If the circular path 2 is outside the core on the outer side of the toroid each turn of the winding passes twice through the area bounded by this path carrying equal currents in opposite directions thus the net current through the area is zero hence eq 1 becomes
B 2 π r = μo x 0 = 0
B = 0
If the circular path 3 is within the core the area bounded by the curve will be threaded by N turns each carrying 1. Thus Current enclosed = NI
Therefore eq 1 becomes
B 2 π r = μo NI
B = μo NI / 2 π r
ELECTROMAGNETIC INDUCTION
The phenomenon in which an Emf is set up in a coil placed in a magnetic field whenever the flux through it is changing is called ELECTROMAGNETIC INDUCTION. If the coil forms a part of a closed circuit the induced Emf cases a current to flow in the circuit. This current is called INDUCED CURRENCY.
The magnitude of induced emf depends upon the rate at which the flux through the coil charges. It also depends on the number of turns on the coil.
The magnetic flux through a circuit can be changed in a number of different ways. By changing the relative position of the coil w.r.t to a magnetic field or current bearing solenoid.
By changing current in the neighbouring coil or by changing current in the coil itself.
By moving a straight conductor in the magnetic field in such a way that it cut the magnetic lines of force.
FLUX LINKAGE
The product of number of turns N and the flux ф through each turn of the coil is called flux linkage i.e.
Flux Linkage = N ф
FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION
A Emf is induced in a coil through which the magnetic flux is changing. The Emf lasts so long as the change of flux is in progress and becomes zero as soon as the flux through the coil becomes constant or zero.
SELF INDUCTION
Consider a coil through which an electric current is flowing. Due to this current magnetic field will be produced which links with the coil itself. If for any reason the current changes the magnetic flux also changes and hence an Emf is induced in the coil this phenomenon is known as SELF INDUCTANCE. In accordance with Lenz Law, the emf posses the change that has induced it and it is therefore known as back emf.
If the current is increasing the back emf opposes the increase. If the current decreasing it opposes the decrease.
The back emf is directly proportional to the rate of change of current. If ΔL change in current Δ t then back emf E is given.
e = L Δl / Δt ——- (I)
Where L = self inductance of the coil.
The measure of the ability of a coil to give rise to a back emf is called the Self inductance. Its value depends on the dimensions of the coil, the number of turns and the permeability of the core material. Its unit is henry.
Henry
The self inductance of a coil is 1 Henry if the current varying through is at the rate of 1 amp/sec, induces a back emf of 1 volt.
If N be the number of turns in the coil and Δ φ be the change of flux in time Δ t then by Faraday’s Law.
Є = -N Δφ / Δt —– (II)
-N Δφ / Δt = – Δl / Δt
N Δφ = L Δl
Δ (Nφ) = Δ (Ll)
Nφ = L1
MUTUAL INDUCTION
Consider two coils close to each other. One coil is connected to a source of emf and the other with a galvanometer. The coil which is connected to the emf is called the primary coil and the other is called secondary coil. Some of the magnetic flux produced by the current in the primary coil is changed the magnetic flux in the secondary coil also changes and hence an emf is induced in the secondary this phenomenon is called mutual induction.
The back emf “ξ” induced in the secondary coil is directly proportional to the rate of change of current Δ1 / Δt in primary coil and is given by
Є2 = -M ΔI / Δt ——– (I)
Where M is the mutual inductance of the pair of coils. Its value depends upon the number of turns of the coil, their cross-sectional area, their closeness and core material. Its unit is Henry.
If N2 be the number of turns in the secondary and Δф / Δt be the rate of change of flux in it then by faraday’s law.
Є2 = -N2 Δφ2 / Δt —— (II)
Comparing 1 and 2
-N2 Δφ2 / Δt = – M Δ1 / Δt
N2 Δφ2 = M Δ1
Δ(N2 φ2) = Δ(M 1)
N2 φ2 = M 1
Non-Inductive Winding
In bridge circuits such as used for resistance measurements self inductance is a nuisance.
When the galvanometer key of bridge is closed the current in the arms of bridge are re-distributed unless the bridge happens to the balanced. When the currents are being re-distributed these are changing and self induction delays the reading of new equilibrium. Thus the galvanometer key thus not corresponds to steady state which the bridge will eventually reach. Its me therefore be misleading.
To minimize their self inductance coils of the bridge and re-resistance boxes are so wound as to setup extremely small magnetic field.
The wire is doubled back on itself before being coiled.
In this type of winding current flows in opposite direction in the double wires and consequently the magnetic field and hence the magnetic flux setup by one wire in neutralized by that due the other wire. Hence self induced emf will not be produced when the current through the circuit changes.
ELECTRICAL MEASURING INSTRUMENTS
CHAPTER – 15
THE MOVING COIL GALVANOMETER
The moving coil galvanometer is a basic electrical instrument. It is used for the detection or measurement of small currents.
Principle
When current flows in a rectangular coil placed in a magnetic field it experience a magnetic torque due to which it rotates through an angle proportional to the current flowing through it.
Construction:
The essential parts of a moving coil galvanometer are
1. A U-shaped permanent magnet with cylindrical concave pole-pieces.
2. A flat coil of thin enamel Insulated wire (usually rectangular)
3. A soft iron cylinder
4. A scalar lamp and scale arrangement.
In suspended type or D Arsonvals galvanometer the flat rectangular coil of thin enamel insulated wire of suitable number of terms wound on a light non-metallic or non-magnetic (brass or aluminum) frame is suspected between the cylindrical concave pole pieces of the permanent U-shaped magnet by a thin phosphor-bronze strip. One end of the wire of the soil is soldered to strip. The other end of the strip is fixed to the frame of the galvanometer and connected to an external terminal. It serves as one current lead. The other end of the wire of the coil is soldered to a loose and soft spiral of wire connected to another external terminal. The soft spiral of wire serves as the other current lead. A soft iron cylinder, coaxial with the pole-pieces is placed within the frame of the coil but quite detached from it and is fixed to the body of the galvanometer. In the space between it and the pole pieces, where the coil moves freely, the soft iron cylinder makes the field stronger and radial so that the magnetic field is always parallel to the plane of the coil. To note the deflection a concave mirror along with lamp and scale arrangement is used.
Working
When a current passes through the galvanometer coil, it experiences a magnetic deflecting torque, which tends to rotate it from its rest position. As the coil rotates it produces a twist in the suspension strip. The coil rotate until the elastic restoring torque due to which the strip does not equalize and cancel the deflecting magnetic torque and then it attains equilibrium and stops rotating further.
i.e. Deflecting torque = Restoring torque
But deflecting torque = BINA Cos α
BINA Cos α = c0
Where B = strength of the magnetic field
I = current in the coil
A = Area of the coil
N = Number of turns in the coil
θ = Angle of twist of the suspension strip
c = torque per unit twist of the suspension strip for the equilibrium
As c/BNA is constant.
In pivoted type or Weston galvanometer the coil instead or being suspended by a strip is pivoted between two jeweled bearings. The restoring torque is provided two hair springs one on either side of the coil and curling on the opposite sense are connected one to each spring. The hairspring thus also serve as current leads to the coil. A light aluminum pointer is fixed to the coil, which moves over a calibrated circular scale with equal divisions, which measures the deflection directly.
Current Sensitivity of a Galvanometer
A galvanometer is said to be sensitive if for a small current the deflection “θ” is sufficiently large. A galvanometer can be made more sensitive if c/BNA is made small. Thus to increase sensitivity “c” may be decreased or B, N and A may be increased “C” can be decreased by increasing the length of suspension wire or by decreasing its can be decreased by increasing the length of suspension wire or by decreasing its diameter, but this process cannot be taken too far < as the suspension must be strong enough to carry the coil. “N” or “A” cannot be increased because it makes the coil heavy. The loss of sensitivity due to the use of fewer turns is however made us by the very high value of the magnetic field employed.
The current sensitivity of a galvanometer is usually defined as the current in microampere required to cause one-millimeter deflection on a scale place 1.0 meter from the mirror of the galvanometer.
THE AMMETER
Ammeter is an instrument, which is used for measuring electric current. A galvanometer can measure small current if its scale is calibrated for the current. For the measurement of large current a bypass resistance called a shunt, of appropriate small value is connected in parallel with the galvanometer coil. This resistance allows the large excess current through itself while a fraction of the current passes through the galvanometer coil. The scale of the instrument is so calibrated that it can measure the main current directly.
Consider a galvanometer “G” whose resistance is “Rg” and which gives full scale deflection when current “Ig” flows through it to convert the galvanometer into an ammeter which can measure a maximum current “I” a shunt “Rs” of appropriate small resistance should be connected in parallel with the galvanometer such that the current “Ig” must flow through the galvanometer coil.
The potential difference “Vg” across the galvanometer is given by
Vg = Ig Rg
The potential difference “Vs” across the shunt is given by
Vs = Is Rs = (I – Ig) Rs
Where Is = I – Ig = current through the shunt.
As “Rg” and “Rs” are connected in parallel to each other therefore potential difference across them will be equal i.e.
Vs = Vg
(I – Ig) Rs = Ig Rg
Rs = Ig Rg / I – Ig
Ammeter is always connected in series with the circuit.
Multi Range Ammeter
Sometimes an ammeter has more than one range, which means that it has as many different shunts as the ranges. The desired range is selected by insertion the proper shunt in position. In one type, one end of each shunt is permanently connected to a common terminal while the other end of each is connected through a range to a second common terminal.
THE VOLTMETER
Voltmeter is an instrument which is used for measuring potential difference between any two points of a current carrying (or between the two terminals of a source of emf). A galvanometer can be used for measuring a very small Potential Difference. If its scale is calibrated for voltage. For the measurement of large potential difference. A high resistance of the order of Kilo-ohms is connected in series with it. This resistance is commonly known as “Multiplier Resistance”.
Consider a galvanometer “G” where resistance is “Rg” and which deflects full scale for the current “Ig” to convert this galvanometer into a voltmeter measuring a Potential difference upto “V” volts. An appropriate high resistance “Rn” must be connected in series with it such that for the potential difference “V” applied between the ends of the above combination. The current “Ig” must flow through the galvanometer. However the total resistance between the terminal a & b is Rn + Rg.
(Rn + Rg+ Ig = V
Rn + Rg = V / Ig
Rn = V / Ig – Rg
Voltmeter is always connected in parallel with the circuit.
Multirange Voltmeter
Sometimes a voltmeter has more than one range, which means it has as many different resistance as the ranges. The desired range is selected by inserting the proper resistance in position.
We have a common terminal marked (+) and as many other terminals as the ranges. In the other type one terminal is common marked (+) while the different range terminals can be connected by a range switch to the other common terminals.
WHEAT STONE BRIDGE
If four resistances R1, R2, R3 and R4 are connected end to end in order to form a closed mesh ABCDA and between one pair of opposite junctions. A and C cell is connected through a key K1 while between the pair of opposite junctions B and D a sensitive galvanometer “G” is connected through another key K2. The circuit so formed is called a “Wheatstone-bridge”.
In the above bridge if the key is closed first, some current flows through the cell and the resistance R1, R2, R3 and R4. If the key K2 is also closed the current will usually be found to flow through the galvanometer indicated by its deflection. However if the resistance R1, R2, R3 and R4 (or at least one of them) are adjusted, a condition can always be attached in which the galvanometer show no deflection at all i.e. no current passes through it. Then the potential difference between B & D must be zero i.e. B & D must be at the same potential. This implies that
P.d. between A and B = p.d. between A and D
OR
V(AB) = V(AD)
P.d. between B and C = p.d. between D and C
OR
V(bt) = V(Dt)
Since no current flows through the galvanometer the current R1 equals that in R2, say II and the current in R3 equals that in R4 say 12
As,
V(AB) = V(AD)
1(1) R(1) = 1(2) R(3) ——– (I)
Also,
V(BC) = V(DC)
1(1) R(2) = 1(2) R(4) ——— (II)
Dividing eq 1 by eq 2
1(1) R(1) / 1(1) R(1) = 1(1) R(1) / 1(1) R(1)
R(1) / R(1) = R(1) / R(1)
Under balanced condition if any three resistance are known then the fourth can be found easily (i.e. wheatstone principle).
The wheatstone principle is used in Meter Bridge, the P.O. box Carey Foster’s Bridge, Callender and Graffite’s Bridge etc.
METER BRIDGE
The Meter Bridge also called slide – Wire Bridge is an instrument based on wheatstone principle. It consists of a long thick copper strip bent twice at right angles. Two small portions are cut off from it near the bends to provide the gaps across which two resistances are known one and an unknown may be connected. Each of the three pieces of the strip is provided with binding screws. A uniform wire (of magnetic or other) one meter long and of fairly high resistance is stretched, along the side a meter scale is connected to the ends of the strip.
For measuring an unknown resistance “X” it is connected in one gap of the Meter Bridge and a standard resistance box “R” is connected in the other gap. A cell and a galvanometer are connected. The jockey “J” is moved along the wire to obtain the balance point D. Under balanced condition if the length of the wire segment. A D toward X is Lx and the length of the wire segment CD towards R is L(R) then their resistances are ρL(R) respectively.
Where ρ = resistance per unit length of the wire.
POST OFFICE BOX [P.O BOX]
Post Office Box is an instrument, which is based on wheatstone principle. It was first introduced for finding resistance of telegraph wires and for fault – findings work in the post and telegraph office that’s why it is called “Post Office Box”. It is more compact and easier to use.
It consists of three sets of resistances P, Q and R. The arms P and Q called the ratio arms, usually consists of three resistances each viz. 10, 100 and 1000 ohms so that any decimal ratio from 1:100 to 100:1 may be used. The third arm “R” is an ordinary set of resistances. The unknown resistances “X” to be measured forms the fourth arm. Introducing the ratios 1:1, 10:1, 100:1 in turn the balance or null position is traced by adjusting “R”. Balance is usually obtained at the ratio 100:1 for some value of “R”. With this value of “R” the value of X can be easily be calculated using relation of Wheatstone bridge i.e.
P/Q = R/X
X = R Q/P
THE OHMMETER
The ohmmeter is a device used for the measurement of resistance. It consists of a sensitive galvanometer “G”, adjustable resister “R” and a torch cell “E” connected in series between two terminals A and B. The unknown resistance “X” to be measured is connected between the terminals A and B. The resistance R is so chosen that when the terminals A and B are short circuited (i.e. X = 0). The galvanometer gives full-scale deflection when no connection is between A and B (i.e. X = ∞). The galvanometer shows zero deflection for the value of X between = and ∞. The deflection is small or large depending on the value of X. The scale of the galvanometer is calibrated with different known values of X and there the circuit serves as an ohmmeter to measure any unknown resistance approx. The scale of the ohmmeter however is not linear.
Using different conditions of R is series and different shunts across the galvanometer worked by range switches, the ohmmeter can be adopted for different accessories for e.g. 1 Ω accuracy in tens of ohms, in hundreds of ohms, in thousands of ohms, in mega ohms etc. Ohmmeter is not a very accurate instrument.
POTENTIOMETER
Potentiometer is device for measuring the p.d (voltage) between two points of a circuit or the e.m.f of a current source. It consists of a uniform wire stretched on a wooden board along a meter scale.
Consider a uniform resistance wire AB of length L and Resistance R, across which is connected to a source of constant EMF (e.g. an accumulator) through a key and a rheostat to adjust and maintain a constant current 1 through it.
As the current flows, the P.d. between A and B = V(AB) = IR
If one terminal of a wire is connected to A while other is moved on the wire AB then instrument acts as a Potential Divider.
To find an unknown EMF of a cell or some other potential difference or the ratio of the emf of two cells consider the circuit. The positive terminals of a cell of unknown EMF “E(N)” and a standard cell of Emf E(N) are connected to the terminal A. The negative terminals of both the cells are joint to the jockey through a two way key and a sensitive galvanometer. Using the two-way key first cell E(N) only is introduced into the galvanometer branch and balanced point C and length L are found for it.
THE AVO-METER
An Avo-meter is an apparatus which is used to measure current, voltage and resistance in other words it is an ampere, volts and ohms. It can measure direct as well as alternative voltage and currents. It consists of a galvanometer with different scales graduated in such a way that all the three quantities can be measured. A selector-cum-range switch is provided. Its has its own battery. A rectifier is also included in the instrument to convert A.C. into D.C. before they pass through the Galvanometer.
Alternating Current
Chapter 16:
INTRODUCTION
In 1974, Sir Augusts de coulomb studied the relationship between localized charges. He carried out experiments using torsion balance. In the basis of his experimental results, he proposed a law known as Coulomb’s Law.
Statement
The electric force of interaction between two point charges is directly proportional to product of their charges and inversely proportional to the square of the distance from their centers.
Mathematical Expression
Consider two points charges q1 and q2 and let “1″ be the separation between them. According to Coulomb’s Law.
F ∞ q1 q2 ——– (i)
And, according to second part
F ∞ 1/r² ——— (ii)
Combining eq (i) and eq (ii), we get
F ∞ q1 q2 / r²
F = K q1 q2 / r²
Where K is the constant of proportionality and its value depends upon the medium between the two charges. In S.I system its value is equal to
K = 8.98 x 10(9) N-m² / c².
Coulomb’s Law can be expressed in terms of permitivity as follows.
F = 1 / 4 π Є . q1 q2 / r²
Where Єo is the permitivity of free space and its value is
Єo = 8.85 x 10(-12) col² / Nm²
if some other medium is used instead of air then
F = 1 / 4 π Єo Єr . q1 q2 / r²
Where Єr is the relative permitivity which is different for different dielectric.
Qs. Define electric field and electric intensity. Find the formula for the electric intensity due to a point charge.
ELECTRIC INTENSITY
DEFINITION
Electric intensity is the force experienced by a unit positive charge due to the presence of a charge body.
EXPLANATION
It is the measure of the strength of the electric field. Electric intensity is a vector quantity and its direction is same as that of force. If the charge is positive then electric intensity is directed from the charge and if the charge is negative, then it is directed towards the charge. The SI unit of electric intensity is Newton per Coulomb or volt per meter.
Mathematical Form
Mathematically electric intensity is given by
E = F / q°
Where E = Electric Intensity
F = Force
q° = small magnitude
ELECTRIC INTENSITY DUE TO A POINT CHARGE
CONSIDERATION
Consider a charged body q that has an electric field all around it. We want to find out electric intensity E at point P. For this purpose, we placed a point charge q1 at that point.
Qs. Define Electric flux. Find the formula for the electric flux due to point charge in a closed sphere.
ELECTRIC FLUX
DEFINITION
The total number of imaginary lines drawn in such a way that the attraction of tangent at any point is same as that of electric field crossing a surface normally is called electric flux or flux on the surface.
OR
The total number of lines of forces crossing a surface normally is called flux on that surface.
Mathematical Form
The flux at a surface is determined by the product of flux density i.e. electric field and the projection of its area perpendicular to the field or by the product of area and component of field normal to the area.
ELECTRIC FLUX DUE TO A POINT CHARGE IN A CLOSED SPHERE
Consider an isolated point charge +q. Th lines of forces from q will spread uniformly in space around it cutting the surface of an imaginary sphere. Now we want to find flux due to point charge. For this purpose, we divide the whole sphere into small patches. Each patch is denoted by ΔA.
Qs. State and prove Guass’s Law
The total electric flux diverging out from a closed surface is equal to the product of the sum of all charges present in that closed surface and 1/ Єo.
Find the Formula for Electric Intensity.
1. Due to Charge Sheet
2. Due to Two Charge Sheets
OR
What are the application of Guass’s Law.
Introduction
Guass’s Law can be used to calculate the electric field only in those cases of charge distribution which are so symmetrical that by proper choice of Guassian surface the flux on it may possibly be evaluated. With the help of guass’s law we can measure the intensity in following cases.
1. Electric Intensity due to charge sheet.
2. Electric Intensity due to two charge sheets.
1. Electric Intensity Due to Charge Sheet.
Consider a charge sheet in which unit positive charges are uniformly distributed.
As we know that charge density is the charge stored per unit area and is denoted by
σ = Q / A
=> Q = σ A
According to Guass’s Law
σ = Q / Є
=> φe = σ A / Є ——– (I)
Now consider a cylindrical shell, which is placed inside the charge sheet. It has three surfaces.
Upper Surface
Curve Surface
Bottom
Electric Intensity Due to Upper Surface
φ1 = E ΔA
φ1 = E ΔA cos θ
But θ = 0° and cos 0° = 1, therefore,
φ1 = E ΔA cos 0°
= φ1 = E ΔA
Electric Intensity Due to Curved Surface
φ2 = E ΔA
φ2 = E ΔA cos θ
Since the angle between the field vector and area vector of all elements of curved surface is 90°, therefore,
φ2 = E ΔA cos 90°
But cos 90° = 0,
φ2 = E ΔA (0)
φ2 = 0
Electric Intensity Due to Lower Surface
φ3 = E ΔA
φ3 = E ΔA cos θ
But 0 = 0° and cos 0°, = 1 therefore,
φ3 = E ΔA cos 0°
=> φ3 = E ΔA
Total Flux
Total Flux is given by
φe = φ1 + φ2 + φ3
=> φe = E ΔA + 0 + E ΔA
=> φe = 2E ΔA
For the whole charge sheet:
=> φe = 2E Σ ΔA
=> φe = 2 EA ———– (II)
Comparing equation (I) and (II)
=> σ A / Єo = 2AE
=> E = σ / 2 Єo
OR
E = σ / 2 Єo
2. Electric Intensity Due to Two Charge Sheets.
As we know that electric intensity due to a charge sheet is
E = σ / 2 Єo
For two sheets E will be,
=> E = σ / 2 Єo + σ / 2 Єo
=> E = σ + σ / 2 Єo
=> E = 2 σ / 2 Єo
=> E = σ / Єo
=> E = σ / Єo i
Qs. Define the term Capacitor and Capacitance of a Capacitor
CAPACITOR
Capacitor is a device which is use to storage charge. A simple capacitor consists of two parallel metallic plates. A plate is connected to the positive terminal of the battery and another plate is connected to the negative terminal of the battery.
CAPACITANCE
Definition
The capacity of a capacitor to store the charge is known as Capacitance.
Mathematical Explanation
If V is the voltage provided to the capacitor and Q is the amount of charge stored in the capacitor, then it is observed that if more is the voltage, then more will be the charge stored in the capacitor.
Mathematically,
Q ∞ V
=> Q = CV
Where C is the capacitance of the capacitor which may be defined as
The ratio of the charge on one of the plate (conductor) to the potential difference between them.
Unit of Capacitance
The unit of Capacitance is Farad. It may be defined as,
If one coulomb charge is stored due to 1 volt potential, then capacitance will be 1 Farad.
Factors on which Capacitance Depends.
The Capacitance of a capacitor depends upon following factors.
1. Cross-sectional area of plate
2. Separation between plates
3. Dielectric
On the basis of dielectric capacitors are classified into different types. For example Electrolyte Capacitor, Paper Capacitor, Meca Capacitor, Oil Capacitor.
Types of Capacitor
There are two main types of Capacitors.
1. Fixed Capacitor
2. Variable Capacitor
1. Fixed Capacitor
Those capacitor whose capacitance is constant are known as Fixed Capacitor. For example Paper Capacitor and Meca Capacitor.
2. Variable Capacitor
Those capacitor whose capacitance is not fixed are known as Variable Capacitor. For example Gang Capacitor.
Find the Formula for the capacitance of Parallel plate capacitor.
PARALLEL PLATE CAPACITOR
Definition
A parallel plate capacitor is a device used to store the charge. It consist of two parallel metallic plates. A plate is connected to the positive terminal of the battery and another plate is connected to the negative terminal of the battery.
These plates are separated by a very small distance compared to the dimension of the plates.
CAPACITANCE OF A PARALLEL PLATE CAPACITOR
Consideration
Consider a parallel plate capacitor in which is the distance between the plates the charge stored is denoted by Q where as potential difference between the plates is V.
Derivation of the Formula
As we known that electric intensity due to two charge sheets is
E = σ / Єo
Since charge density is given by
σ = Q / A
Substituting the value in above equation
E = Q / A Єo ——— (I)
According to the definition of electrical potential.
ΔV = E Δ
Substituting the value of E from eq (I) in above equation
Q = C x Qd / A Єo
C = A Єo / d
If some other medium is used instead of air then,
C = A Єo Єr / d
Find the formula for equivalent capacitance when,
1. Capacitors are connected in Series
2. Capacitors are connected in Parallel
Introduction
Capacitors of some fixed values are used in a circuit. The capacitance of the desired value can however be obtained by suitable combination of capacitor. Capacitors can be combined in parallel, series or both.
WHEN CAPACITORS ARE CONNECTED IN SERIES OR SERIES COMBINATION
Consideration
Consider three capacitors having capacitance C1, C2, C3 connected in a series. These capacitors can be replace by an equivalent capacitor having capacitance Ce. When a cell is connected across the ends of system then a charge Q is transferred across the plates of capacitors. A charge upon one plate always attracts upon the other plate with a charge equal in magnitude and opposite in sign.
Let V be the potential difference across the combination. The potential difference across the individual capacitor is Vab, Vbc, Vcd.
Derivation of the Formula
As we know that in case of series combination the charge across the individual capacitor remains constant, where as potential difference varies such that the potential difference V is the sum of potential difference applied across individual capacitor.
Vad = Vab + Vbc + Vcd ———— (I)
We know that,
Q = CV
V = Q / C
Therfore,
Vad = Q / Ce
Vab = Q / C1
Vbc = Q / C2
Vcd = Q / C3
Substituting the values of Vad, Vab, Vbc and Vcd in eq (I)
Q/Ce = Q/C1 + Q/C2 + Q/C3
Q/Ce = Q [1/C1 + 1/C2 + 1/C3]
1/Ce = 1/C1 + 1/C2 + 1/C3
Conclusion
The reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitance.
WHEN CAPACITORS ARE CONNECTED IN PARALLEL OR PARALLEL COMBINATION
Consideration
Consider three capacitors having C1, C2, C3 capacitance respectively are connected in parallel. We can replace them by an equivalent capacitor having capacitance Ce. A charge q given to a point divide it self reside on the plates of individual capacitor as Q1, Q2, Q3 respectively.
Derivation of the Formula
As we known that in case of parallel combination the potential difference across each capacitor is that of the source where as the charge across each capacitor varies, therefore the total charge Q is given by
Q = Q1 + Q2 + Q3 ——— (I)
As we known that,
Q = CV
Therefore,
Q = Ce Vab
=> Q1 = C1 Vab
=> Q2 = C2 Vab
=> Q3 = C3 Vab
Substituting the values of Q1, Q2, Q3 in eq (I)
Ce Vab = C1 Vab + C2 Vab + C3 Vab
Ce Vab + Vab (C1 + C2 + C3)
Ce = C1 + C2 + C3
Conclusion
The equivalent capacitance is equal to sum of the individual capacitances.
Qs. Define electric potential and absolute potential.
ELECTRIC POTENTIAL
Definition
Electric Potential is the amount of work done in order to bring a unit positive charge from one point to another point against the direction of electric field.
Explanation
In order to bring a unit positive charge from one point to another point, we have to do some work. This work is stored in the form of potential energy. This potential energy per unit charge is known as Electric Potential.
Mathematical Form
Electric Potential is denoted by V. It is defined as the potential energy per unit charge. Therefore mathematically,
ΔV = Δu / q ——- (I)
We know that
Δu = F. Δr
But E = F/q => F = Eq, therefore,
Δu = EqΔr
Substituting the value in eq (I)
ΔV = EqΔr / q
=> ΔV = E. Δr
Where,
ΔV = Electric Potential
E = Electric Intensity
Δr – Displacement
From above expression we know that
Electric Potential is the dot product of electric intensity and displacement.
It is a scalar quantity and its unit is Volt.
ABSOLUTE POTENTIAL
Definition
Absolute potential is the amount of work done in order to bring a unit positive charge from one point to infinite distance against the direction of magnetic field.
ELECTRIC POTENTIAL
Electric Potential is the amount of work done in order to move a unit positive charge from one point to another against the direction of electric.
Consideration
Consider a unit positive charge placed in a uniform electric field. We have to displace it from point O to point N. For this purpose we have to do some work. This work is known as electric potential. In order to determine electric potential from point O to point N we divide the whole distance into small equal patches because in a long distance intensity does not remain constant. This patch is denoted by Δr.
CURRENT ELECTRICITY
CHAPTER – 13
Qs. Define Charge and Current.
CHARGE
Definition
Flow of electron is known as Charge.
It is denoted by Q.
Unit
Its unit is Coulomb.
1 Coulomb = 10(-6) μcoulomb
1 coulomb = 10 (-3) mili coulomb
1 coulomb = 10(-9) neno coulomb
CURRENT
Definition
The flow of charge per unit time is known as Current.
It is denoted by I.
Unit
The unit of current is coulomb/sec or Ampere.
AMPERE
If one coulomb charge passes through the conductor in 1 second then the current is 1 Ampere.
Mathematical Form
Mathematically,
I = Q/t
Qs. State and Explain Ohm’s Law.
OHM’S LAW
Introduction
A German scientist George Simon Ohm studied the relationship between voltage, current and resistance. On the basis of his experimental results, he proposed a law which is known as Ohm’s Law.
Statement
Ohm’s Law to metallic conductors can be stated as
The current through a conductor is directly proportional to the potential difference between the ends of the conductor provided that physical conditions are kept constant.
It can also be stated as
The ratio between voltage and current remains constant, if the physical conditions are kept constant.
Mathematical Form
Mathematically,
V ∞ I
V = IR
R = V/I
Where R is the constant of proportionality known as resistance of the conductor. Its unit is volt per ampere (Volt/Ampere) or Ohm (Ω).
Ohm (Ω)
If 1 ampere current passes through the conductor due to 1 volt potential difference then the resistance of conductor is 1 Ohm.
Resistance
Opposition offered in the flow of current.
Graphical Representation.
When graph is plotted between current and potential differences then straight line is obtained.
Limitations of the Law
Ohm’s Law is valid only for metallic resistance at a given temperature and for steady currents.
Qs. Define the term Resistivity or Coefficient of Resistor.
RESISTIVITY OR COEFFICIENT OF RESISTOR
Definition
It is the resistance of a unit conductor whose cross-sectional area is 1 sqm.
Unit
Its unit is Ohm meter.
Mathematical Form
The resistance of any conductor depends upon the following factors.
1. Length of the conductor
2. Cross-sectional area of the conductor.
3. Material of the conductor.
The resistance of the conductor is directly proportional to the length of the conductor and inversely proportional to the cross-sectional area.
Mathematically,
R ∞ L ——– (I)
R α 1/A —— (II)
Combining eq (I) and (II)
R α L/A
=> R = ρL/A
Where ρ is the constant of proportionality known as Resistivity or Coefficient of resistance.
ρ = RA/L
Qs. Explain the effect of temperature on resistance or temperature coefficient of resistance.
EFFECT OF TEMPERATURE ON RESISTANCE
It is observed that if we increase the temperature then resistance of a conductor will increase.
Consideration
Let Ro be the initial resistance of a conductor at 4°C. If we increase the temperature from t1°C to t2°C, then resistance will increase. This increment in resistance is denoted by ΔR. The increment in resistance depends upon the following two factors.
1. Original Resistance (Ro)
2. Difference in temperature Δt.
Mathematical Verification
The increment in resistance is directly proportional to the original resistance and temperature difference.
Mathematically,
ΔR ∞ Ro —– (I)
ΔR ∞ Δt —– (II)
Combining eq (I) and eq (II) we get
ΔR ∞ RoΔt
=> ΔR = αRoΔt
Where α is the temperature coefficient of resistance. It is defined as
It is the increment in resistance per unit resistance per degree rise in temperature.
Its unit is 1/°C or °C. If RT is the total resistance, then
RT = Ro + ΔR
=> RT = Ro + αRo Δt
=> RT = Ro (1 + αΔt)
As we know that resistance is directly proportional to resistivity therefore,
ρT = ρo (1 + αΔt)
Qs. Define the term Power Decipation in Resistor.
POWER DECIPATION IN RESISTORS
Definition
When current flows in a conductor then a part of electrical energy appears in the form of heat energy which is known as Power Decipation in Resistor.
Units
Its unit is Joule per second (J/s). Most commonly used unit is Kwh.
1 Kwh = 36 x 10(5) Joules
Mathematical Form
Since,
P = Electrical Work / Time
Electrical Work = QV —— (I)
This electrical work produces heat energy in the resistor.
P = QV / t
P = Q / t . V
But,
I = Q / t
P = VI
From Ohm’s Law
V = IR
P = IIR
P = I2R
OR,
P = 12R2 / R
=> P = V2 / R
As we know that,
Energy = Power x time
=> E = P x t
=> E = Vit
=> E = I2Rt
And,
E = V2 / R . t
Qs. Define and explain Electromotive Force.
ELECTROMOTIVE FORCE
Definition
It is the terminal voltage difference when no current draws from a cell or a battery.
OR
Work done per coulomb on the charges.
It is denoted by E.
Unit
Electromotive force or simply e.m.f is a scalar quantity it has the same dimension as that of voltage, therefore its unit is volt.
Explanation
When an electric current passes through a resistor, it dissipates energy, which is transformed into heat energy. Thus to sustain a current in conductor some source of energy is needed so that it could continuously supply power equal to that which is dissipated as heat in the resistor. The strength of this source is called Electromotive Force.
Consideration
Let consider a simple circuit in which a resistor “R” is connected by leads of negligible resistance to the terminals of a battery. The battery is made up of some electrolyte and electrode for the production of e.m.f and hence when this current flows from battery, it encounters some resistance by the electrolyte present in two electrodes. This resistance is known as internal resistance “r” of the battery.
Mathematical Form
According to Ohm’s Law
V = IR
I = V / R
Or,
I = E / R + r
Where E is e.m.f and r is internal resistance
=> E = IR + Ir
E = V + Ir
MAGNETISM AND ELECTRO-MAGNETISM
CHAPTER-14
MAGNETIC FIELD DUE TO CURRENT
It was discovered by Oersted that when current masses through a conductor, magnetic field is produced. This field is known as “Magnetic Field of Induction” and is denoted bu “B”.
Ampere found that when two current carrying conductors are near each other, they experience force at each other. If the current is in the same direction the force is attractive and if the current is in opposite direction.
When electric charges are at rest they exert electrostatic force of attraction or repulsion on each other. When the charges are in motion they exert electric as well as magnetic force on each other because and isolated moving positive and negative charge create both electric and magnetic field.
MAGNETIC FIELD
Magnetic Field is a space or region around a magnet or current carrying coil of wire where its effect can be felt by small compass needle. Magnetic field of induction can be visualized by magnetic lines of induction.
A line of induction is an endless curve, which can be traced by a compass needle.
MAGNETIC FLUX AND FLUX DENSITY
The total number of magnetic lines of induction passing through a surface is called magnetic flux.
DETERMINING THE CHARGE TO MASS RATIO OF AN ELECTRON
The charge to mass into of an electron was determined by Sir J.J. Thomson by an apparatus which consists of a highly evacuated pear shaped glass pulls into which several metallic electrodes are sealed.
Electrons are produced by heating a tungsten flament F by passing a current through it. The electrons moving sideways are also directed towards the screen by applying negative potential on a hollow cylinder C open on both the sides surrounding the filament. Electrons are accelerated by applying positive potential to discs A and B. If V be the total total P.d between the disc Band the filament F taken then Kinetic Energy.
The beam strikes the screen coated with zinc sulphide after passing through the middle of the two horizontal moetal P’P and a spot of the light produced at O on the screen where the beam strikes and its position is noted.
A magnetic field of induction B is produced in between the plate directed into the paper. The magnetic field is produced by two identical current carrying coils placed on either side of the tube at the position of plates.
The force due to the magnetic field on the moving electron makes them move in a curved path and the light spot shifts from O to O on the screen there from of magnetic field acts as centripetal force
e. V B = mv2 / r
e/m = V/Br ——– I
e/m can be computed if the radius r and the expression of the circular path are in which the beam moves in the field region is determined. The radius r is calculated from the shift of the light spot i.e. r = 3.
A better method of determined V is as under. An electric field E is produced between the plates by applying suitable potential difference to exert a force “Be” on the electron opposite to that due to the magnetic field.
The potential diff. VI is so adjusted that two fields neutralize each other effects and the spot come back to its initial position O. Thus each other effects and the spot come back to its initial position O. Thus
Ee = Be V
Or
V = E/B —– (II)
Where E = V1 / d
d = distance between the plates.
Putting the value of V from eq 2 in 1
e/m = E/B2r
e/m = K75888 x 10(11) e/kg
AMPERE’S LAW
According to this law the sum of the product of the tangential component of the magnetic field of indaction and te length of an element of a closed curve taken in a magnetic field is μo times the current which passes through that area bounded by the curve.
Consider a long straight wire carrying a current 1 in the direction. The lines of force are concentric circles with their common centre on the wire. From these circles consider a circle of radius r. The magnitude of the magnetic field at all points on this circle and inside the circle is same.
Biot and Savart experimentally found that the magnitude of the field depends directly on twice the current and inversely proportional to the distance r from the conductor.
SOLENOIDAL FIELD
A solenoid is a coil of an insulated copper wire wound on a circular cylinder with closed turns. When current passes through it, magnetic field is produced with is uniform and strong inside the solenoid while outside it the field is negligibly weak.
Consider a solenoid through which the current 1 is passing in order to determine the magnetic field of induction B at any point inside the solenoid imagine a closed path “abcda” on the form of a rectangular. The rectangular is divided into four elements of length L1, L2, L3, L4. L1 is along the axis inside the solenoid and L3 is far from the solenoid.
By applying amperes circuital law
B L1 + B. L2 + L2 + B. L3 + B. L4 = μo x current enclosed —– (I)
Since B. L1 is parallel inside the solenoid
B. L1 = BL4 cos 0 = BL4
The field is very weak outside the solenoid is very weak and therefore it can be negnected thus
B. L3 = 0
As B is perpendicular to L2 and L4 inside the solenoid therefore
B. L2 = BL2 cos 90 = 0
B. L4 = BL4 cos 90 = 0
substitute the above values is eq 1
B. L1 + O + 0 + 0 = μo x current closed
B. L1 = μo x current enclosed ——- (II)
If there are n turns per unit length of the solenoid and each turn carries a current I will be “n L1I”
TOROIDAL FIELD
A Toroid or a circular solenoid is a coil of insulated copper wire wound on a circular core with close turn. When the current passes through the toroid, magnetic field is produced which is strong enough inside while outside it is almost zero.
Consider a toroid that consists of N closely packed turns that carry a current I. Imagine a circular curve of concentric the core.
It is evident form of the symmetry at all points of the curve must have the same magnitude an should be tangential to the curve at all points. Divide the circle into small elements each of length ΔL is so small that B and ΔL are parallel to each other.
By amperes law
Σ B : ΔL = μo x current enclosed
ΣB ΔL Cos 0 = μo x current enclosed
ΣB ΔL = μo x current enclosed
BΣ ΔL = μo x current enclosed
Σ ΔL = 2 π r
B 2 π r = μo x current enclosed ——– (I)
Cases
If the circular path 1 is outside the core on the inner side of the toroid if enclose no current. Thus eq 1 become
B 2 π r = μo x 0 = 0
B = 0
If the circular path 2 is outside the core on the outer side of the toroid each turn of the winding passes twice through the area bounded by this path carrying equal currents in opposite directions thus the net current through the area is zero hence eq 1 becomes
B 2 π r = μo x 0 = 0
B = 0
If the circular path 3 is within the core the area bounded by the curve will be threaded by N turns each carrying 1. Thus Current enclosed = NI
Therefore eq 1 becomes
B 2 π r = μo NI
B = μo NI / 2 π r
ELECTROMAGNETIC INDUCTION
The phenomenon in which an Emf is set up in a coil placed in a magnetic field whenever the flux through it is changing is called ELECTROMAGNETIC INDUCTION. If the coil forms a part of a closed circuit the induced Emf cases a current to flow in the circuit. This current is called INDUCED CURRENCY.
The magnitude of induced emf depends upon the rate at which the flux through the coil charges. It also depends on the number of turns on the coil.
The magnetic flux through a circuit can be changed in a number of different ways. By changing the relative position of the coil w.r.t to a magnetic field or current bearing solenoid.
By changing current in the neighbouring coil or by changing current in the coil itself.
By moving a straight conductor in the magnetic field in such a way that it cut the magnetic lines of force.
FLUX LINKAGE
The product of number of turns N and the flux ф through each turn of the coil is called flux linkage i.e.
Flux Linkage = N ф
FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION
A Emf is induced in a coil through which the magnetic flux is changing. The Emf lasts so long as the change of flux is in progress and becomes zero as soon as the flux through the coil becomes constant or zero.
SELF INDUCTION
Consider a coil through which an electric current is flowing. Due to this current magnetic field will be produced which links with the coil itself. If for any reason the current changes the magnetic flux also changes and hence an Emf is induced in the coil this phenomenon is known as SELF INDUCTANCE. In accordance with Lenz Law, the emf posses the change that has induced it and it is therefore known as back emf.
If the current is increasing the back emf opposes the increase. If the current decreasing it opposes the decrease.
The back emf is directly proportional to the rate of change of current. If ΔL change in current Δ t then back emf E is given.
e = L Δl / Δt ——- (I)
Where L = self inductance of the coil.
The measure of the ability of a coil to give rise to a back emf is called the Self inductance. Its value depends on the dimensions of the coil, the number of turns and the permeability of the core material. Its unit is henry.
Henry
The self inductance of a coil is 1 Henry if the current varying through is at the rate of 1 amp/sec, induces a back emf of 1 volt.
If N be the number of turns in the coil and Δ φ be the change of flux in time Δ t then by Faraday’s Law.
Є = -N Δφ / Δt —– (II)
-N Δφ / Δt = – Δl / Δt
N Δφ = L Δl
Δ (Nφ) = Δ (Ll)
Nφ = L1
MUTUAL INDUCTION
Consider two coils close to each other. One coil is connected to a source of emf and the other with a galvanometer. The coil which is connected to the emf is called the primary coil and the other is called secondary coil. Some of the magnetic flux produced by the current in the primary coil is changed the magnetic flux in the secondary coil also changes and hence an emf is induced in the secondary this phenomenon is called mutual induction.
The back emf “ξ” induced in the secondary coil is directly proportional to the rate of change of current Δ1 / Δt in primary coil and is given by
Є2 = -M ΔI / Δt ——– (I)
Where M is the mutual inductance of the pair of coils. Its value depends upon the number of turns of the coil, their cross-sectional area, their closeness and core material. Its unit is Henry.
If N2 be the number of turns in the secondary and Δф / Δt be the rate of change of flux in it then by faraday’s law.
Є2 = -N2 Δφ2 / Δt —— (II)
Comparing 1 and 2
-N2 Δφ2 / Δt = – M Δ1 / Δt
N2 Δφ2 = M Δ1
Δ(N2 φ2) = Δ(M 1)
N2 φ2 = M 1
Non-Inductive Winding
In bridge circuits such as used for resistance measurements self inductance is a nuisance.
When the galvanometer key of bridge is closed the current in the arms of bridge are re-distributed unless the bridge happens to the balanced. When the currents are being re-distributed these are changing and self induction delays the reading of new equilibrium. Thus the galvanometer key thus not corresponds to steady state which the bridge will eventually reach. Its me therefore be misleading.
To minimize their self inductance coils of the bridge and re-resistance boxes are so wound as to setup extremely small magnetic field.
The wire is doubled back on itself before being coiled.
In this type of winding current flows in opposite direction in the double wires and consequently the magnetic field and hence the magnetic flux setup by one wire in neutralized by that due the other wire. Hence self induced emf will not be produced when the current through the circuit changes.
ELECTRICAL MEASURING INSTRUMENTS
CHAPTER – 15
THE MOVING COIL GALVANOMETER
The moving coil galvanometer is a basic electrical instrument. It is used for the detection or measurement of small currents.
Principle
When current flows in a rectangular coil placed in a magnetic field it experience a magnetic torque due to which it rotates through an angle proportional to the current flowing through it.
Construction:
The essential parts of a moving coil galvanometer are
1. A U-shaped permanent magnet with cylindrical concave pole-pieces.
2. A flat coil of thin enamel Insulated wire (usually rectangular)
3. A soft iron cylinder
4. A scalar lamp and scale arrangement.
In suspended type or D Arsonvals galvanometer the flat rectangular coil of thin enamel insulated wire of suitable number of terms wound on a light non-metallic or non-magnetic (brass or aluminum) frame is suspected between the cylindrical concave pole pieces of the permanent U-shaped magnet by a thin phosphor-bronze strip. One end of the wire of the soil is soldered to strip. The other end of the strip is fixed to the frame of the galvanometer and connected to an external terminal. It serves as one current lead. The other end of the wire of the coil is soldered to a loose and soft spiral of wire connected to another external terminal. The soft spiral of wire serves as the other current lead. A soft iron cylinder, coaxial with the pole-pieces is placed within the frame of the coil but quite detached from it and is fixed to the body of the galvanometer. In the space between it and the pole pieces, where the coil moves freely, the soft iron cylinder makes the field stronger and radial so that the magnetic field is always parallel to the plane of the coil. To note the deflection a concave mirror along with lamp and scale arrangement is used.
Working
When a current passes through the galvanometer coil, it experiences a magnetic deflecting torque, which tends to rotate it from its rest position. As the coil rotates it produces a twist in the suspension strip. The coil rotate until the elastic restoring torque due to which the strip does not equalize and cancel the deflecting magnetic torque and then it attains equilibrium and stops rotating further.
i.e. Deflecting torque = Restoring torque
But deflecting torque = BINA Cos α
BINA Cos α = c0
Where B = strength of the magnetic field
I = current in the coil
A = Area of the coil
N = Number of turns in the coil
θ = Angle of twist of the suspension strip
c = torque per unit twist of the suspension strip for the equilibrium
As c/BNA is constant.
In pivoted type or Weston galvanometer the coil instead or being suspended by a strip is pivoted between two jeweled bearings. The restoring torque is provided two hair springs one on either side of the coil and curling on the opposite sense are connected one to each spring. The hairspring thus also serve as current leads to the coil. A light aluminum pointer is fixed to the coil, which moves over a calibrated circular scale with equal divisions, which measures the deflection directly.
Current Sensitivity of a Galvanometer
A galvanometer is said to be sensitive if for a small current the deflection “θ” is sufficiently large. A galvanometer can be made more sensitive if c/BNA is made small. Thus to increase sensitivity “c” may be decreased or B, N and A may be increased “C” can be decreased by increasing the length of suspension wire or by decreasing its can be decreased by increasing the length of suspension wire or by decreasing its diameter, but this process cannot be taken too far < as the suspension must be strong enough to carry the coil. “N” or “A” cannot be increased because it makes the coil heavy. The loss of sensitivity due to the use of fewer turns is however made us by the very high value of the magnetic field employed.
The current sensitivity of a galvanometer is usually defined as the current in microampere required to cause one-millimeter deflection on a scale place 1.0 meter from the mirror of the galvanometer.
THE AMMETER
Ammeter is an instrument, which is used for measuring electric current. A galvanometer can measure small current if its scale is calibrated for the current. For the measurement of large current a bypass resistance called a shunt, of appropriate small value is connected in parallel with the galvanometer coil. This resistance allows the large excess current through itself while a fraction of the current passes through the galvanometer coil. The scale of the instrument is so calibrated that it can measure the main current directly.
Consider a galvanometer “G” whose resistance is “Rg” and which gives full scale deflection when current “Ig” flows through it to convert the galvanometer into an ammeter which can measure a maximum current “I” a shunt “Rs” of appropriate small resistance should be connected in parallel with the galvanometer such that the current “Ig” must flow through the galvanometer coil.
The potential difference “Vg” across the galvanometer is given by
Vg = Ig Rg
The potential difference “Vs” across the shunt is given by
Vs = Is Rs = (I – Ig) Rs
Where Is = I – Ig = current through the shunt.
As “Rg” and “Rs” are connected in parallel to each other therefore potential difference across them will be equal i.e.
Vs = Vg
(I – Ig) Rs = Ig Rg
Rs = Ig Rg / I – Ig
Ammeter is always connected in series with the circuit.
Multi Range Ammeter
Sometimes an ammeter has more than one range, which means that it has as many different shunts as the ranges. The desired range is selected by insertion the proper shunt in position. In one type, one end of each shunt is permanently connected to a common terminal while the other end of each is connected through a range to a second common terminal.
THE VOLTMETER
Voltmeter is an instrument which is used for measuring potential difference between any two points of a current carrying (or between the two terminals of a source of emf). A galvanometer can be used for measuring a very small Potential Difference. If its scale is calibrated for voltage. For the measurement of large potential difference. A high resistance of the order of Kilo-ohms is connected in series with it. This resistance is commonly known as “Multiplier Resistance”.
Consider a galvanometer “G” where resistance is “Rg” and which deflects full scale for the current “Ig” to convert this galvanometer into a voltmeter measuring a Potential difference upto “V” volts. An appropriate high resistance “Rn” must be connected in series with it such that for the potential difference “V” applied between the ends of the above combination. The current “Ig” must flow through the galvanometer. However the total resistance between the terminal a & b is Rn + Rg.
(Rn + Rg+ Ig = V
Rn + Rg = V / Ig
Rn = V / Ig – Rg
Voltmeter is always connected in parallel with the circuit.
Multirange Voltmeter
Sometimes a voltmeter has more than one range, which means it has as many different resistance as the ranges. The desired range is selected by inserting the proper resistance in position.
We have a common terminal marked (+) and as many other terminals as the ranges. In the other type one terminal is common marked (+) while the different range terminals can be connected by a range switch to the other common terminals.
WHEAT STONE BRIDGE
If four resistances R1, R2, R3 and R4 are connected end to end in order to form a closed mesh ABCDA and between one pair of opposite junctions. A and C cell is connected through a key K1 while between the pair of opposite junctions B and D a sensitive galvanometer “G” is connected through another key K2. The circuit so formed is called a “Wheatstone-bridge”.
In the above bridge if the key is closed first, some current flows through the cell and the resistance R1, R2, R3 and R4. If the key K2 is also closed the current will usually be found to flow through the galvanometer indicated by its deflection. However if the resistance R1, R2, R3 and R4 (or at least one of them) are adjusted, a condition can always be attached in which the galvanometer show no deflection at all i.e. no current passes through it. Then the potential difference between B & D must be zero i.e. B & D must be at the same potential. This implies that
P.d. between A and B = p.d. between A and D
OR
V(AB) = V(AD)
P.d. between B and C = p.d. between D and C
OR
V(bt) = V(Dt)
Since no current flows through the galvanometer the current R1 equals that in R2, say II and the current in R3 equals that in R4 say 12
As,
V(AB) = V(AD)
1(1) R(1) = 1(2) R(3) ——– (I)
Also,
V(BC) = V(DC)
1(1) R(2) = 1(2) R(4) ——— (II)
Dividing eq 1 by eq 2
1(1) R(1) / 1(1) R(1) = 1(1) R(1) / 1(1) R(1)
R(1) / R(1) = R(1) / R(1)
Under balanced condition if any three resistance are known then the fourth can be found easily (i.e. wheatstone principle).
The wheatstone principle is used in Meter Bridge, the P.O. box Carey Foster’s Bridge, Callender and Graffite’s Bridge etc.
METER BRIDGE
The Meter Bridge also called slide – Wire Bridge is an instrument based on wheatstone principle. It consists of a long thick copper strip bent twice at right angles. Two small portions are cut off from it near the bends to provide the gaps across which two resistances are known one and an unknown may be connected. Each of the three pieces of the strip is provided with binding screws. A uniform wire (of magnetic or other) one meter long and of fairly high resistance is stretched, along the side a meter scale is connected to the ends of the strip.
For measuring an unknown resistance “X” it is connected in one gap of the Meter Bridge and a standard resistance box “R” is connected in the other gap. A cell and a galvanometer are connected. The jockey “J” is moved along the wire to obtain the balance point D. Under balanced condition if the length of the wire segment. A D toward X is Lx and the length of the wire segment CD towards R is L(R) then their resistances are ρL(R) respectively.
Where ρ = resistance per unit length of the wire.
POST OFFICE BOX [P.O BOX]
Post Office Box is an instrument, which is based on wheatstone principle. It was first introduced for finding resistance of telegraph wires and for fault – findings work in the post and telegraph office that’s why it is called “Post Office Box”. It is more compact and easier to use.
It consists of three sets of resistances P, Q and R. The arms P and Q called the ratio arms, usually consists of three resistances each viz. 10, 100 and 1000 ohms so that any decimal ratio from 1:100 to 100:1 may be used. The third arm “R” is an ordinary set of resistances. The unknown resistances “X” to be measured forms the fourth arm. Introducing the ratios 1:1, 10:1, 100:1 in turn the balance or null position is traced by adjusting “R”. Balance is usually obtained at the ratio 100:1 for some value of “R”. With this value of “R” the value of X can be easily be calculated using relation of Wheatstone bridge i.e.
P/Q = R/X
X = R Q/P
THE OHMMETER
The ohmmeter is a device used for the measurement of resistance. It consists of a sensitive galvanometer “G”, adjustable resister “R” and a torch cell “E” connected in series between two terminals A and B. The unknown resistance “X” to be measured is connected between the terminals A and B. The resistance R is so chosen that when the terminals A and B are short circuited (i.e. X = 0). The galvanometer gives full-scale deflection when no connection is between A and B (i.e. X = ∞). The galvanometer shows zero deflection for the value of X between = and ∞. The deflection is small or large depending on the value of X. The scale of the galvanometer is calibrated with different known values of X and there the circuit serves as an ohmmeter to measure any unknown resistance approx. The scale of the ohmmeter however is not linear.
Using different conditions of R is series and different shunts across the galvanometer worked by range switches, the ohmmeter can be adopted for different accessories for e.g. 1 Ω accuracy in tens of ohms, in hundreds of ohms, in thousands of ohms, in mega ohms etc. Ohmmeter is not a very accurate instrument.
POTENTIOMETER
Potentiometer is device for measuring the p.d (voltage) between two points of a circuit or the e.m.f of a current source. It consists of a uniform wire stretched on a wooden board along a meter scale.
Consider a uniform resistance wire AB of length L and Resistance R, across which is connected to a source of constant EMF (e.g. an accumulator) through a key and a rheostat to adjust and maintain a constant current 1 through it.
As the current flows, the P.d. between A and B = V(AB) = IR
If one terminal of a wire is connected to A while other is moved on the wire AB then instrument acts as a Potential Divider.
To find an unknown EMF of a cell or some other potential difference or the ratio of the emf of two cells consider the circuit. The positive terminals of a cell of unknown EMF “E(N)” and a standard cell of Emf E(N) are connected to the terminal A. The negative terminals of both the cells are joint to the jockey through a two way key and a sensitive galvanometer. Using the two-way key first cell E(N) only is introduced into the galvanometer branch and balanced point C and length L are found for it.
THE AVO-METER
An Avo-meter is an apparatus which is used to measure current, voltage and resistance in other words it is an ampere, volts and ohms. It can measure direct as well as alternative voltage and currents. It consists of a galvanometer with different scales graduated in such a way that all the three quantities can be measured. A selector-cum-range switch is provided. Its has its own battery. A rectifier is also included in the instrument to convert A.C. into D.C. before they pass through the Galvanometer.
Alternating Current
Chapter 16:
ELECTRONICS
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| Electronics is the branch of physics which deals with development of electron-emitting devices, there use and control of electron flow in electrical circuits. Electronics also deals with semiconductors, diode, rectifiers etc. | ||
p-TYPE
SUBSTANCE
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| If a trivalent element from the IIIrd group such as Gallium (Ga) or Indium (In) is added to pure crystals of germanium (Ge) or silicone (Si), three electrons of impurity form covalent bonds with three atoms of (Ge) or (Si), while there exist a vacancy for an electron in the fourth bond. This vacant space is called Hole. This hole behaves like a positive charge and can move in the structure of substance. Such a substance is called a p-type substance. | ||
n-TYPE
SUBSTANCE
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| If a pentavalent element from the Vth Group such as Antimony (Sb) is added to pure geranium (Ge) or silicone (Si), then four electrons of (Sb) will form covalent bonds with four (Ge) or (Si) atoms. The fifth electron of 'Sb' is free to move which makes (Ge) or (Si) a good conductor. This type of material is called n-type substance. | ||
RECTIFIER
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| A
rectifier is a device which is used to convert alternating current (AC)
into direct current (DC). PN-junction diode is used as a rectifier. |
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RECTIFICATION
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| The process of converting alternating current into direct current is called rectification. | ||
FORWARD
BIASING
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| when n-type end of pn-junction is connected to negative terminal and p-type end with positive terminal of a (DC) supply, then the height of potential barrier reduces and provides easy flow of electric charge that is pn-junction conducts electricity. In this condition pn-junction is said to be Forward Biased. | ||
REVERSE
BIASING
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| When p-type end of pn-junction is connected to the negative terminal and n-type end with positive terminal of a (DC) supply. The height of potential barrier increases to maximum and the flow of electric charge across the junction will become zero. In this condition a pn-junction diode is called Reverse Biasing. | ||
DOPING
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Addition of an element of group IIIrd-A or Vth-A to Ge or Si crystals to
convert them into semiconductor substance (p-type or n-type) is called Doping.
Normally impurity is in very small quantity. There are two types of impurities
that are added to geranium or silicon:
DONOR IMPURITY
ACCEPTOR IMPURITY | ||
INTRODUCTION
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| A pn - junction diode is an electronic device formed from a p-type and an n-type substance semiconductor. A semiconductor diode has the property of one way conduction i.e. it allows electric current to flow in only one direction. | ||
FABRICATION
OF pn-JUNCTION
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| A
pn-junction is fabricated by placing a small amount of indium on a plate
or wafer of n -type germanium. Indium on heating at 550oC melts
and diffuses through a small part of the n-type germanium. Indium being
a p-type impurity, converts the part of the n-type germanium to p-type material.
Thus a junction is formed between p-type section and an n-type section of
germanium. A brass-base is used to fix the pn-junction to which leads are attached as shown: |
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| The whole apparatus is sealed in a glass tube or a metallic tube. | ||
WORKING
OF pn-JUNCTION DIODE
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| As
we know that a p-type substance has excess of mobile positive charge or
holes and n-type substance has an excess of negative charge or electrons,
the electrons from n-type and holes from p-type sections flow across the
junction and combine. In this way a layer of positive charges is formed
on the n-type and a layer of negative charges on p-type material. Due to induction of these layers a potential barrier is now developed across the junction and further flow of charges is prevented from one side to the other. | ||
TRANSISTORS
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| A three terminal semiconductor electronic device is called transistor. Transistors are widely used in electronic appliances such as computers, radio, audio video equipment, bio medical instrument etc. | ||
CONSTRUCTION
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| A transistor is a three layer semiconductor which consist a very thin central layer of one type of semiconductor material sandwiched between two relatively thick layer of second type. | ||
The no. of electrons emitted per second i.e. photo current is proportional
to the intensity of incident light.
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