Prasanna

Students can Download Chapter 13 Probability Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 13 Probability

Introduction
In this chapter we study the concept of conditional probability, multiplication rule of probability and independence of events, Baye’s theorem, Probability distribution and its mean and variance, Bernoulli Trials, Binomial Distribution, and its mean and variance.

A. Basic Concepts
I. Conditional Probability
Let A and B are two events associated with the sample space of a random experiment. The probability of occurrence of the event A that the event B has already happened is called Conditional Probability of A given B, denoted by P(A/B).

Properties:

1. Let A and B be events of a sample space S of an experiment, then P(S/A) = 1, P(A/A) = 1
2. If A and B are two events of a sample space S and E is an event of S such that P(E) ≠ 0, then P((A U B)/E) = P(A/E) + P(B/E) – P((A ∩ B)/E)
3. P(A’/B) = 1 – P(A/B).

II. Multiplication Theorem
If A and B be two events associated with a random experiment, then

1. P(A ∩ B) = P(B) × P(A/B), if P(B) ≠ 0
2. P(A ∩ B) = P(A) × P(B/A), if P(A) ≠ 0

P(A ∩ B ∩ C) = P(A) × P(B / A) × P(C /(A ∩ B)).

III. Independent Events
Two events are said to be independent if the probability of occurrence of any one of the event does not affect the occurrence of the other.

1. P(A/B) = P(A)
2. P(A ∩ B) = P(A) × P(B)
3. P(A ∩ B ∩ C) = P(A) × P(B) × P(C)
4. Two events associated with a random experiment cannot be both mutually exclusive and independent.
5. If P(A ∩ B) ≠ P(A) × P(B), the A and B are dependent events.
6. If A and B are independent events, then
   • A and \(\bar{B}\) are independent events,
   • \(\bar{A}\) and \(\bar{B}\) are independent events.

IV. Theorem of total probability
Let {E₁, E₂,…., E_n}be a partition of the sample space S, and suppose that each of the events E₁, E₂, …., E_n has nonzero probability of occurrence. Let A be any event associated with S, then
P(A) = P(E₁)P(A/E₁) + P(E₂)P(A/E₂) +…….+ P(E_n)P(A/E_n)
= \(\sum_{i=1}^{n}\)P(E_i)P(A/E_i).

V. Baye’s Theorem
If E₁, E₂,…., E_n are n non-empty events which constitute a partition of sample space S, then

VI. Probability Distribution
The Probability Distribution of a random variable X is the system of numbers

Where the real numbers x₁, x₂,…….., x_n are the possible values of the random variable X and p₁, p₂,……, P_n is the probability of the each possible values of the random variable X.
P₁ + P₂+…….+P_n = 1 and 0 ≤ p_t ≤ 1.

1. The mean of the above Probability Distribution is denoted by µ, is also called Expectation of X.
   ie; Mean = µ = E(X) = \(\sum_{i=1}^{n}\)x_ip_i
2. The Variance of the above Probability Distribution is denoted by σ²,
   ie; Variance = σ² = E(X²) – [E(X)]²

VII. Bernoulli Trial
Trials of a random experiment are called Bernoulli trial, if it satisfies the following conditions:

1. There should be a finite number of trials.
2. The trials should be independent.
3. Each trial has exactly two outcomes: success or failure.
4. The probability of success remains the same in each trial.

VIII. Binomial Distribution
Binomial Distribution, denote by B(n, p) is given (p + q)ⁿ where p represents probability of success, q represent probability of failure and n represents the number of trials. The probability of x success is
P(X = x) = nC_xq^{n – x}p^x.

1. Mean = np
2. Variance = npq
3. Standard Deviation = \(\sqrt{n p q}\).

Students can Download Chapter 2 Reproductive Health Notes, Plus Two Zoology Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Zoology Notes Chapter 2 Reproductive Health

Reproductive health – problems and strategies
The Govt of India has started family planning (1951)and ‘Reproductive and Child Health Care (RCH)
programmes’ that helps
1. To create awareness among people about various reproduction related aspects.

2. Introduction of sex education in schools to avoid myths and misconceptions about sex-related aspects.

3. The information about reproductive organs, adolescence and related changes, safe and hygienic sexual practices, sexually transmitted diseases (STD), AIDS, etc., would help adolescent age group to lead a reproductively healthy life.

4. The information of available birth control options, care of pregnant mothers, post-natal care of the mother and child, importance of breastfeeding, equal opportunities for the male and the female child, etc., are the important components build up a socially responsible and healthy society.

5. The ban on amniocentesis for sex-determination helps to legally check increasing female foeticides, massive child immunisation, etc., are some programmes.

Population Explosion And Birth Control
According to the census report of 2001

Population increased due to decreased

1. Maternal mortality rate (MMR)
2. Infant mortality rate (IMR)

According to the 2001 census report, the population growth rate was 1.7 per cent.
Marriageable age raised to 18 years for females and 21 years for males. Government was taken measures to check this population growth rate by the contraceptive methods.

An ideal contraceptive should be user-friendly, easily available, effective and reversible with no or least side-effects.. Natural/Traditional, Barrier, lUDs, Oral contraceptives, Injectables, Implants and Surgical methods.

Natural Methods:
Periodic abstinence – Here the couples avoid or abstain from coitus from day 10 to 17 of the menstrual cycle.

Withdrawal or coitus interruptus is another method in which the male partner withdraws his penis from the vagina just before ejaculation so as to avoid insemination.

Lactational amenorrhea This method is based on the fact that ovulation and therefore the cycle do not occur during the period of intense lactation following parturition. So chances of conception are almost nil.

Barrier Method:
Condoms (Nirodh’) are barriers made of thin rubber/ latex sheath that are used to cover the penis in the male or Wagina and cervix in the female, just before coitus. This can prevent conception.

Use of condoms have additional benefit of protecting the user from STDs and AIDS. Both the male and the female condoms are disposable.

Diaphragms, cervical caps and vaults are also barriers made of rubber that are inserted into the female reproductive tract to cover the cervix during coitus. So it helps to prevent conception by blocking the entry of sperms through the cervix.

Intra Uterine Devices (lUDs):
IUDs are ideal contraceptives for the females. These devices are inserted by doctors or expert nurses in the uterus through vagina.

Two types are copper releasing lUDs (CuT, Cu7, Multiload 375) and the hormone releasing lUDs (Progestasert, LNG-20)

IUDs increase phagocytosis of sperms within the uterus and the Cu ions released suppress sperm motility and the fertilising capacity of sperms.

The hormone releasing lUDs make the uterus unsuitable for implantation It is one of most widely accepted methods of contraception in India.

Oral contraceptives(pills) contain either progestogens or progestogen-estrogen combinations.

Pills have to be taken daily for a period of 21 days starting within the first five days of menstrual cycle. After a gap of 7 days (during which menstruation occurs) it has to be repeated in the same pattern to prevent conception.

It helps to inhibit ovulation and implantation as well as to prevent entry of sperms, eg-Saheli.

Saheli-a new oral contraceptive for the females-was developed by scientists at Central Drug Research Institute (CDRI) in Lucknow.

Administration of progestogens or progestogen-estrogen combinations or lUDs within 72 hours of coitus have been found to be very effective as emergency contraceptives to avoid possible pregnancy due to rape or casual unprotected intercourse.

Surgical Methods (Sterilisation):

In male it is called ‘vasectomy’ a small part of the vas deferens is removed or tied up through a small incision on the scrotum.

In female, ‘tubectomy’ a small part of the fallopian tube is removed or tied up through a small incision in the abdomen or through vagina.

Medical Termination Of Pregnancy (MTP)
Government of India legalised MTP in 1971 to avoid its misuse. So it helps to check indiscriminate and illegal female foeticides which are reported to be high in India.

It is important to avoid unwanted pregnancies either due to casual unprotected intercourse or failure of the contraceptive used during coitus or rapes.

MTPs are also essential in the cases where continuation of the pregnancy could be harmful or even fatal either to the mother or to the foetus or both.

MTPs are considered relatively safe during the first trimester, i.e., upto 12 weeks of pregnancy.

Sexually Transmitted Diseases (STD_s)
Diseases or infections which are transmitted through sexual inter course are called sexually transmitted diseases (STD) or venereal diseases (VD) or reproductive tract infections (RTI).
Common STDs.

Early symptoms of most of these are minor and include itching, fluid discharge, slight pain, swellings, etc., in the genital region.

Hepatitis-B and HIV are transmitted by sharing of injection needles, surgicaljnstruments, etc. with infected persons and transfusion of blood, from an infected mother to the foetus.

All diseases are completely curable except hepatitis-B, genital herpes, and HIV infections.
The important steps to control STDs are given below.

Infertility
In the case of infertility, corrections are not possible, the couples could be assisted to have children through certain special techniques commonly known as assisted reproductive technologies (ART).

Students can Download Chapter 12 Linear Programming Notes, Plus Two Maths Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Maths Notes Chapter 12 Linear Programming

Introduction
A special class of optmisation problems such as finding maximum profit, minimum cost, or minimum use of resources, etc, is Linear Programming Problems. In this chapter we study some linear programming problems and their solutions graphically.

A. Basic Concepts
A linear Programming Problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called the objective function) subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are non-negative.
A few important LPP are;

• Diet Problem.
• Manufacturing Problem.
• Transportation Problem.

1. The common region determined by the constraints including the non-negative constraints x ≥ 0, y ≥ 0 of a LPP is called the feasible region.

2. Points within and on the boundary of the feasible region represents feasible solution of the constraints. Any point outside the feasible region is an infeasible solution.

3. Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.

I. Corner point Method

1. Find the feasible region of the LPP and determine its corner points (vertices).
2. Evaluate the objective function Z = ax + by at each corner points. Let M and m be the maxjmum and minimum values at these points.
3. If the feasible region is bounded, M, and m respectively are the maximum and minimum values of the objective function.
4. If the feasible region is unbounded, then
   • M is the maximum value of the objective function, if the open half-plane determined by ax + by > M has no points in common with the feasible region. Otherwise the objective function no maximum value.
   • m is the minimum value of the objective function, if the open half-plane determined by ax + by < m has no point in common with the feasible region. Otherwise, the objective function has no minimum value.

Students can Download Chapter 3 Current Electricity Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 3 Current Electricity

Introduction
In the present chapter, we shall study some of the basic laws concerning steady electric currents.

Electric Current
Rate of flow of electric charge is called electric current. or

Electric Currents In Conductors

• Conductors: Free electrons are found in conductors. The electric current in conductors is due to the flow of electrons.
• Electrolytes: The current in electrolyte is due to flow of ions.
• Semiconductor: The current in semiconductor is due to flow of both holes and electrons.

Ohm’s Law
At constant temperature, the current through a conductor is directly proportional to the potential difference between its ends.
V α I (or)

where R is constant, called resistance of materials.

1. Resistance of a material:
Factors Affecting Resistance of Resistor:
For a given material resistance is directly proportional to the length and inversely proportional to the area of cross-section.
R ∝ \(\frac{\mathrm{L}}{\mathrm{A}}\)

where ρ is the constant of proportionality and is called resistivity of material.

Resistivity (coefficient of specific resistance) of a Material:
The resistance per unit length for unit area of cross-section will be a constant and this constant is known as the resistivity of the material. The resistivity or coefficient of specific resistance is defined as the resistance offered by a resistor of unit length and unit area of cross-section.

Resistivity is a scalar quantity and its unit is Ω-m.

Conductance and conductivity:
The reciprocal of resistance is called conductance and the reciprocal of resistivity is called conductivity. The SI unit of conductance is seimen and that for conductivity is seimen per meter. The unit of conductance can also be expressed as Ω^-1.

Current density: Current per unit area is called current density
current density j = \(\frac{I}{A}\)
Vector form

Mathematical expression of Ohm’s law in terms of j and E:
Considers conductor of length i. Let V’ be the potential difference between the two ends of a conductor.
According to ohms law
We know V= IR

This potential difference produces an electric field E in the conductor. The p.d. across the conductor also can be written as
V = El _____(2)
Comparing (1) and (2), we get
El = jρl
E = jρ
The above relation can be written in vector form as

where σ is called conductivity of the material.

Drift Of Electrons And The Origin Of Resistivity
Random thermal motion of electrons in a metal:
Every metal has a large number of free electrons. Which are in a state of random motion within the conductor. The average thermal speed of the free electrons in random motion is of the order of 10⁵m/s.

Does random thermal motion produce any current? The directions of thermal motion are so randomly distributed. Hence the average thermal velocity of the electrons is zero. Hence current due to thermal motion is zero.

(a) Drift Velocity (vd):
The average velocity acquired by an electron under the applied electric field is called drift velocity.

Explanation: When a voltage is applied across a conductor, an electric filed is developed. Due to this electric field electrons are accelerated. But while moving they collide with atoms, lose their energy and are slowed down. This acceleration and collision are repeated through the motion. Hence electrons move with a constant average velocity. This constant average velocity is called drift velocity.

(b) Relaxation time (τ):
Relaxation time is the average of the time between two successive collisions of the free electrons with atoms.

(c) Expression for drift velocity:
Let V be the potential difference across the ends of a conductor. This potential difference makes an electric field E. Under the influence of electric field E, each free electron experiences a Coulomb force.
F = -eE
or ma = -eE
a = \(\frac{-e E}{m}\) _____(1)
Due to this acceleration, the free electron acquires an additional velocity. A metal contains a large number of electrons.
For first electron, additional velocity acquired in a time τ,
v₁ = u₁ + aτ₁
where u₁ is the thermal velocity and τ is the relaxation time.
Similarly the net velocity of second, third,……electron

v₂ = u₂ + aτ₂
v₃ = u₃ + aτ₃
v_n = u_n + aτ_n
∴ Average velocity of all the ‘n’ electrons will be

V_av = 0 + aτ (∴ average thermal velocity of electron is zero)
where τ = \(\frac{\tau_{1}+\tau_{2}+\ldots \ldots \ldots+\tau_{n}}{n}\)
where V_av is the average velocity of electron under an external field. This average velocity is called drift velocity.
ie. drift velocity V_d = aτ _____(2)

(d) Relation between electric current and drift speed:
Consider a conductor of cross-sectional area A. Let n be the number of electrons per unit volume. When a voltage is applied across a conductor, an electric filed is developed. Let v_d be the drift velocity of electron due to this field.
Total volume passed in unit time = Av_d
Total number of electrons in this volume = Av_dn
Total charge flowing in unit time=Av_dne
But charge flowing per unit time is called current I ie. current, I = Av_dne
I = neAv_d
Now current density J can be written as
J = nev_d (J= I/A)
Deduction of Ohm’s law: (Vector form)
We know current density
J = nv_de

1. Mobility: Mobility is defined as the magnitude of the drift velocity per unit electric field.

Limitations Of Ohm’s Law
Certajn materials do not obey Ohm’s law. The deviations of Ohm’s law are of the following types.
1. V stops to be proportional to I.

Metal shows this type behavior. When current through metal becomes large, more heat is produced. Hence resistance of metal increases. Due to increase in resistance the V-I graph becomes nonlinear. This nonlinear variation is shown by solid line in the above graph.

2. Diode shows this type behavior. We get different values of current for same negative and positive voltages.

3. This type behavior is shown by materials like GaAs. there is more than one value of V for the same current I.

Note: The materials which donot obey ohms law are mainly used in electronics.

Resistivity Of Various Materials
The materials are classified as conductors, semi conductors, and insulators according to their resistivities. Commercially produced resistors are of two types.

1. Wire bound resistors
2. Carbon resistors

1. Wire bound resistor:
Wire wound resistors are made by winding the wires of an alloy.
Eg: Manganin, Constantan, Nichrome.

2. Carbon resistors:
Resistors in the higher range are made mostly from carbon. Carbon resistors are compact. Carbon resistors are small in size. Hence their values are given using a colourcode.

(i) Colourcode of resistors:
The resistance value of commercially available resistors are usually indicated by certain standard colour coding.

The resistors have a set of coloured rings on it. Their significance is indicated in the table The first two bands from the end indicated the first two significant digits and the third band indicates the decimal multiplier. The last metallic band indicates the tolerance.
Value of colours:

Illustration:

The colour code indicated in the given sample is Red, Red, Red with a silver ring at the right end. Then the value of given resistance is 22 × 10² ±10%.

Temperature Dependence Of Resistivity:
The resistivity of a material is found to be dependent on the temperature. The resistivity of a metallic conductor is approximately given by,
ρ_T = ρ_o[1 + α(T – T_o)]
where ρ_T is the resistivity at a temperature T and ρ_o is the resistivity at temperature T_o. α is called the temperature coefficient of resistivity.
Variation of resistivity in metals:

The temperature coefficient (α) of metal is positive. Which means resistivity of metal increases with temperature. The variation of resistivity of copper is as shown in above figure.
Eg: Silver, copper, nichrome, etc.
Variation of resistivity in semi conductor:

The temperature coefficient (α)of semiconductor is negative. Which means that resistivity decreases with increase in temperature. The variation of resistivity with temperature for a semiconductor is shown in above figure.
Eg: Carbon,Germanium,silicon
Variation of resistivity in standard resistors:

standard resistors, the variation of resistivity will be very little with temperatures. The variation of resistivity with temperature for standard resistors is show above.
Eg: Manganin and constantan.
Explanation for the variation of resistivity:
The resistivity of a material is given by

The above equation shows that, resistivity depends inversely on number density and relaxation time τ.
Metals:
Number density in metal does not change with temperature. But average speed of electrons increases. Hence frequency of collision increases. The increase in frequency of collision decreases the relaxation time τ. Hence the resistivity of metal increases with temperature.

Insulators and semiconductors:
For insulators and semiconductors, the number density n increases with temperature. Hence resistivity decreases with temperature.

Electrical Energy, Power

Consider two points A and B in a conductor. Let V_A and V_B be the potentials at A and B respectively. The potential At A is greater than that B and difference in potential is V.

If ∆Q charge flows from A to B in time ∆t. The potential energy of charge will be decreased. The decrease in potential energy due to charge flow from A to B,
= V_A∆Q – V_B∆Q
= (V_A – V_B)∆Q
= V∆Q (V_A – V_B = V)
This decrease in PE appeared as KE of flowing charges. But we know, the kinetic energy of charge carriers do not increase due to the collisions with atoms. During collisions, the kinetic energy gained by the charge carriers is shared with the atoms.

Hence the atoms vibrate more vigorously ie. The conductor heats up. According to conservation of energy, heat developed in between A and B in time ∆t
∆H = decrease in potential energy
∆H = V∆Q
∆H = VI∆t (∵ ∆Q=I∆t)
\(\frac{\Delta \mathrm{H}}{\Delta \mathrm{t}}\) = VI
Rate of workdone is power, ie

using Ohm’s law V = IR

It is this power which heats up the conductor.
Power transmission:
The electric power from the electric power station is transmitted with high voltage. When voltage increases, the current decreases. Hence heat loss decreases very much.

Combination Of Resistors – Series And Parallel Combination
Resistors in series:
Consider three resistors R₁, R₂ and R₃ connected in series and a pd of V is applied across it.

In the circuit shown above the rate of flow of charge through each resistor will be same i.e. in series combination current through each resistor will be the same. However, the pd across each resistor are different and can be obtained using ohms law.
pd across the first resistor V₁ = I R₁
pd across the second resistor V₂, = I R₂
pd across the third resistor V₃ = I R₃
If V is the effective potential drop and R is the effective resistance then effective pd across the combination is V = IR
Total pd across the combination = the sum pd across each resistor, V = V₁ + V₂ + V₃
Substituting the values of pds we get IR = IR₁ + IR₂ + IR₃
Eliminating I from all the terms on both sides we get

Thus the effective resistance of series combination of a number of resistors is equal to the sum of resistances of individual resistors.

1. Resistors in parallel:
Consider three resistors R₁, R₂ and R₃ connected in parallel across a pd of V volt. Since all the resistors are connected across same terminals, pd across all the resistors are equal.

As the value of resistors are different current will be different in each resistor and is given by Ohm’s law
Current through the first resistor
I₁ = \(\frac{V}{R_{1}}\)
Current through the second resistor
I₂ = \(\frac{V}{R_{2}}\)
Current through the third resistor
I₃ = \(\frac{V}{R_{3}}\)
Total current through the combination is
I = \(\frac{V}{R}\), where R is the effective resistance of parallel combination.
Total current through the combination = the sum of current through each resistor
I = I₁ + I₂ + I₃
Substituting the values of current we get

Eliminating V from all terms on both sides of the equations, we get

Thus in parallel combination reciprocal of the effective resistance is equal to the sum of reciprocal of individual resistances. The effective resistance in a parallel combination will be smaller than the value of smallest resistance.

Cells, Emf, Internal Resistance
Electrolytic cell:
Electrolytic cell is a simple device to maintain a steady current in an electric circuit. A cell has two electrodes. They are immersed in an electrolytic solution.

E.M.F:
E.M.F. is the potential difference between the positive and negative electrodes in an open circuit. ie. when no current is flowing through the cell.

Voltage:
Voltage is the potential difference between the positive and negative electrodes, when current is flowing through it.

Internal resistance of cell:
Electrolyte offers a finite resistance to the current flow. This resistance is called internal resistance (r).

Relation between ε, V and internal drop:
Consider a circuit in which cell of emfs is connected to resistance R. Let r be the internal resistance and I be the current following the circuit.

According to ohms law,
current flowing through the circuit

V is the voltage across the resistor called terminal voltage. Ir is potential difference across internal resistance called internal drop.

Cells In Series And Parallel

Consider two cells in series. Let ε₁, r₁ be the emf and internal resistance of first cell. Similarly ε₂, r₂ be the emf and internal resistance of second cell. Let I be the current in this circuit.
From the figure, the P.d between A and B
V_A – V_B = ε₁ – Ir₁ _____(1)
Similarly P.d between B and C
V_B – V_C = ε₂ – Ir₂ ______(2)
Hence, P.d between the terminals A and C
V_AC = V_A – V_C = V_A – V_B + V_B – V_C
V_AC = [V_A – V_B] + [V_B – V_C]
when we substitute eqn. (1) and (2) in the above equation.
V_AC = ε₁ – Ir₁ + ε₂ – Ir₂ V_AC = (ε₁ – ε₂) – I(r₁ + r₂)
V_AC = ε_eq – Ir_eq
where ε_eq = ε₁ + ε₂, and r_eq = r<sub1 + r₂
The rule of series combination:

1. The equivalent emf of a series combination of n cells is the sum of their individual emf.
2. The equivalent internal resistance of a series com-bination of n cells is just the sum of their internal resistances.

Cells in parallel:

Consider two cells connected in parallel as shown in figure. ε₁, r₁ be the emf and internal resistance of first cell and ε₂, r₂ be the emf and internal resistance of second cell. Let I₁ and I₂ be the current leaving the positive electrodes of the cells.
Total current flowing from the cells is the sum of I₁ and I₂.
ie. I = I₁ + I₂ ______(1)
Let V_B1 and V_B2 be the potential at B₁ and B₂ respectively. Considering the first cell, P.d between B₁ and B₂.

Considering the second cell, P.d between B₁ and B₂

Substituting the values I₁ and I₂ in eq.(1), we get


If we replace the combination by a single cell between B₁ and B₂, of emf and ε_eq and internal resistance r_eq, we have
V = ε_eq – Ir_eq ______(3)
The eq(2) and eq(3) should be same.

The above equation can be put in a simpler way.

If there are n cells of emf ε₁, ε₂,………..ε_n and internal
resistance r₁, r₂………..r_n respectively, connected in parallel.

Kirchoff’s Rules
1. First law (Junction rule): The total current entering the junction is equal to the total current leaving the junction.
Explanation:

Consider a junction ‘O’. Let I₁ and I₂ be the incoming currents and I₁, I₄ and I₅ be the outgoing currents.
According to Kirchoff’s first law,

2. Second law (loop rule): In any closed circuit the algebraic sum of the product of the current and resistance in each branch of the circuit is equal to the netjemf in that branch.

OR

Total emf in a closed circuit is equal to sum of voltage drops

Explanation: Consider a circuit consisting of two cells of emf E₁ and E₂ with resistances R₁, R₂ and R₃ as shown in figure. Current is flowing as shown in figure
Applying the second law to the closed circuit ABCDE₁A.
-I₃R₃ + E₁ + -I₁R₁ = 0
Similarly for the closed loop ABCDE₂A.
-I₂R₂ + -I₃R₃ + E₂ = 0
For the closed loop AE₂DE₁A
-I₁R₁ + I₂R₂ + -E₂ + E₁ = 0
Note:

• Voltage drop in the direction of current is taken as negative (and vice versa).
• emf is taken as positive, if we go -ve to +ve terminal (and vice versa)

Wheatstone’s Bridge
Four resistances P, Q, R, and S are connected as shown in figure. Voltage ‘V’ is applied in between A and C. Let I₁, I₂, I₃ and I₄ be the four currents passing through P, R, Q, and S respectively.

Working:
The voltage across R
When key is closed, current flows in different branches as shown in figure. Under this situation
The voltage across P, V_AB = I₁P
The voltage across Q, V_BC = I₃Q __(1)
The voltage across R, V_AD = I₂R
The voltage across S, V_DC = I₄S
The value of R is adjusted to get zero deflection in galvanometer. Underthis condition,
I₁ = I₃ and I₂ = I₄ _____(2)
Using Kirchoffs second law in loopABDA and BCDB, weget
V_AB = V_AD ______(3)
and V_BC = V_DC _______(4)
Substituting the values from eq(1) into (3) and (4), we get
I₁P = I₂R ______(5)
and I₃Q = I₄S _____(6)
Dividing Eq(5) by Eq(6)

[since I₁ = I₃ and I₂ = I₄]
This is called Wheatstone condition.

Meter Bridge
Uses: Meter Bridge is used to measure unknown resistance.
Principle: It works on the principle of Wheatstone bridge condition (P/Q=R/S).

Circuit details:
Unknown resistance X’ is connected in between A and B. Known resistance (box) is connected in between B and C. Voltage is applied between A and C. A100cm wire is connected between A and C. Let r be the resistance per unit length. Jockey is connected to ‘B’ through galvanometer.
Working: A suitable resistance R is taken in the box. The position of jockey is adjusted to get zero deflection.
If ‘l’ is the balancing length from A, using Wheatstone’s condition,

knowing R and l, we can find X (resistance of wire)
Resistivity: Resistivity of unknown resistance (wire) can be found from the formula
\(\rho=\frac{\pi r^{2} X}{l}\)
Where r (the radius of wire) is measured using screw gauge. l (the length of wire) is measured using meter scale
Note: Meter bridge is most sensitive when all the four resistors are of the same order

Potentiometer
(a) Comparison of e.m.f of two cells using potentiometer:

Principle: Potential difference between two points of a current carrying conductor (having uniform thickness) is directly proportional to the length of the wire between two points.

Circuit details: A battery (B₁), Rheostat and key are connected in between A and B. This circuit is called primary circuit. Positive end of E₁ and E₂ are connected to A and other ends are connected to a two way key. Jockey is connected to a two key through galvanometer. This circuit is called secondary circuit.

Working and theory: Key in primary circuit is closed and then E₁ is put into the circuit and balancing length l₁ is found out.
Then, E₁ α l₁ ______(1)
Similarly, E₂ is put into the circuit and balancing length (l₂ ) is found out.
Then, E₂ α l₂ _______(2)
Dividing Eq(1) by Eq(2),
\(\frac{E_{1}}{E_{2}}=\frac{l_{1}}{l_{2}}\) _____(3)

(b) Measurement of internal resistance using potentiometer:
Principle: Potential difference between two points of a current carrying conductor (having uniform thickness) is directly proportional to the length of the wire between two points.

Circuit details: Battery B₁, Rheostat and key K1 are connected in between A and B. This circuit is called primary.

In the secondary circuit a battery E having internal resistance ‘r’ is connected . A resistance box (R) is connected across the battery through a key (K^{2). Jockey is connected to battery through galvanometer.}

Working and theory : The key (K₁) in the primary circuit is closed and the key is the secondary (K₂) is open. Jockey is moved to get zero deflection in galvanometer. The balancing length l₁ (from A) is found out.
Then we can write.
E₁ α l₁ _____(1)
Key K₂ is put in the circuit, corresponding balancing length (l₂) is found out. Let V be the applied voltage, then
V₁ α l₁ _____(2)
‘V’ is the voltage across resistance box.
Current through resistance box
ie, voltage across resistance,
V = \(\frac{E R}{(R+r)}\) ____(3)
Substituting eq (3) in eq (2),
\(\frac{E R}{(R+r)} \alpha l_{2}\) ____(4)
Dividing eq (1) by eq (4),

Question 1.
Why is potentiometer superior to voltmeter in measuring the e.m.f of a cell?
Answer:
Voltmeter takes some current while measuring emf. So actual emf is reduced. But potentiometer does not take current at null point and hence measures actual e.m.f. Hence potentiometer is more accurate than voltmeter.

Students can Download Chapter 6 Electromagnetic Induction Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 6 Electromagnetic Induction

Introduction
In this chapter we are going to discuss the laws governing electromagnetic induction; how energy can be stored in a coil, generation of ac, the relation between voltage and current in various circuit components and finally the working of transformer.

The Experiments Of Faraday And Henry
Faraday and Henry conducted a series of experiments to develop principles of electro magnetic induction.
Experiment – 1

Connect a coil to a galvanometer G as shown in the figure. When the north pole of a bar magnet is pushed towards the coil, galvanometer shows deflection. The deflection indicates that a current is produced in the coil.

The galvanometer does not show any deflection when the magnet is held stationary. When the magnet is pulled away from the coil, the galvanometer shows deflection in the opposite direction.

Conclusion of experiment 1:
The relative motion between magnet and coil produces an electric current in the first coil.
Experiment 2

Connect a coil C₁ to a galvanometer G. Take another coil C₂ and connect it with a battery. A steady current in the coil produces a steady magnetic field. When the coil C₂ is moved towards the coil C₁, the galvanometer shows a deflection. This deflection indicates that the electric current is induced in the coil G.

When the coil C₂ is moved away, the galvanometer shows a deflection in the opposite direction. When the coil C₂ is kept fixed, no deflection is produced in the coil C₁.
Conclusion of Experiment – 2:
The relative motion between two coils induces an electric current in the first coil.
Experiment – 3

In this experiment coil C₁ is connected to galvanometer G. The second coil C₂ is connected to a battery through a key K.

When the key K is pressed, the galvanometer shows a deflection. If the key is held pressed continuously, there is no deflection in the galvanometer. When the key is released, a momentary deflection is observed again, (but in opposite direction).

Conclusion of experiment – 3:
The change in current in second coil induces a current in the first coil.

Magnetic Flux

Magnetic flux through a plane of area A placed in uniform magnetic field B can be written as

Φ = BAcosθ

Faraday’S Law Of Induction
Faraday’s law of electromagnetic induction states that the magnitude of the induced emf in a circuit is equal to the time rate of change of magnetic flux through the circuit.
Mathematically, the induced emf is given by

If the coil contain N turns, the total induced emf is given by,

Lenz’S Law And Conservation Of Energy
Lenz’s law:
Lenz’s law states that the direction emf (or current) is such that it opposes the change in magnetic flux which produces it,
Mathematically the Lenz’s law can be written as

The negative sign represents the effect of Lenz’s law. The magnitude of the induced emf is given by the Faraday’s law. But Lenz’s law gives the direction induced emf.
Lenz’s law is an accordance with the law of conservation of energy.

When the north pole of the magnet is moved towards the coil, the side of the coil facing north pole becomes north as shown in above figure. (Current is produced in the coil and flows in anticlockwise direction).

So work has to be done to move a magnet against this repulsion. This work is converted into electrical energy.

When the north pole of the magnet is moved away from the coil, the end of the coil facing the north pole acquires south polarity. So work has to be done to overcome the attraction. This work is converted into electrical energy. This electrical energy is dissipated as heat produced by the induced current.

Motional Electromotive Force

Consider a rectangular frame MSRN in which the conductor PQ is free to move as shown in figure. The straight conductor PQ is moved towards the left with a constant velocity v perpendicular to a uniform magnetic field B. PQRS forms a closed circuit enclosing an area that change as PQ moves. Let the length RQ = x and RS = I.
The magnetic flux Φ linked with loop PQRS will be BIx.
Since x is changing with time the rate of change of flux Φ will induce an e.m.f. given by

Energy Consideration: A Quantitative Study
Let ‘r’ be the resistance of arm PQ. Consider the resistance of arm QR, RS, and SP as zero. When the arm is moved,
The current produced in the loop,

This current flows through the arm PQ. The arm PQ is placed in a magnetic field. Hence force acting on the arm,

Substituting eq. (1) in eq. (2)

If this arm is pulling with a constant velocity v, the power required for motion P = Fv

The external agent that does this work is mechanical. Where does this mechanical energy go?
This energy is dissipated as heat. The power dissipated by Joule law,

From eq. (3) and (4), we can understand that the workdone to pull the conductor is converted into heat energy in conductor.
Relation between induced charge and magnetic flux:
We know

so the above equation can be written as

We also know magnitude of induced emf

Comparing (1) and (2), we get

Eddy Currents
Whenever the magnetic flux linked with a metal block changes, induced currents are produced. The induced currents flow in a closed paths. Such currents are called eddy currents.
Experiment to Demonstrate Eddy Currents
Experiment -1

Allow a rectangular metal sheet to oscillate in between the pole pieces of strong magnet as shown in figure. When the plate oscillates, the magnetic flux associated with the plate changes. This will induce eddy currents in the plate. Due to this eddy current the rectangular metal sheet comes to rest quickly.
Experiment – 2

Make rectangular slots on the copper plate. These slots will reduce area of plate. Allow this copper plate to oscillate in between magnets. Due to this decrease in area, the eddy current is also decreased. Hence the plate swings more freely.

Some important applications of Eddy Currents:
1. Magnetic braking in trains:
Strong electromagnets are situated above the rails. When the electromagnets are activated, eddy currents induced in the rails. This eddy current will oppose the motion of the train.

2. Electromagnetic damping:
Certain galvanometers have a core of metallic material. When the coil oscillates, the eddy currents are generated in the core. This eddy current opposes the motion and brings the coil to rest quickly.

3. Induction furnace:
Induction furnace can be used to melt metals. A high frequency alternating current is passed through a coil. The metal to be melted is placed in side the coil. The eddy currents generated in the metals produce heat, that melt it.

4. Electric power meters:
The metal disc in the electric power meter (analogue type) rotates due to the eddy currents. This rotation can be used to measure power consumption.

Inductance
An electric current can be induced in a coil by two methods:

1. Mutual induction
2. Self induction

1. Mutual inductance:
Mutual induction:
The phenomenon of production of an opposing e.m.f. in a circuit due to the change in current or magnetic flux linked with a neighboring circuit is called mutual induction.
Explanation

Consider two coils P and S. P is connected to a battery and key. S is connected to a galvanometer. When the key is pressed, a change in magnetic flux is produced in the primary.

This flux is passed through S. So an e.m.f. is produced in the ‘S’. Thus we get a deflection in galvanometer. similarly, when key is opened, the galvanometer shows a deflection in the opposite direction.

Mutual inductance or coefficient of mutual induction:
The flux linked with the secondary coil is directly proportional to the current in the primary
i.e. Φ α I Or Φ = MI
Where M is called coefficient of mutual induction or mutual inductance.
If I = 1, M = Φ
Hence mutual inductance of two coils is numerically equal to the magnetic flux linked with one coil, when unit current flows through the other.

(i) Mutual inductance of two coils:
Expression for mutual inductance:
Consider a solenoid (air core) of cross sectional area A and number of turns per unit length n. Another coil of total number of turns N is closely wound over the first coil. Let I be the current flow through the primary. Flux density of the first coil B= µ₀nI
Flux linked with second coil, Φ = BAN
Φ = µ₀nIAN _____(1)
But we know Φ = MI _______(2)
From eq(1) and eq(2), we get
∴ MI = µ₀nIAN
M = µ₀nAN
If the solenoid is covered over core of relative permeability µ_r
then M = µ_rµ₀nAN

Relation between induced e.m.f. and coefficient of mutual inductance:
Relation between induced e.m.f and mutual inductance
We know induced e.m.f

2. Self inductance:
Self-induction
The phenomenon of production of an induced e.m.f in a circuit when the current through it changes is known as self- induction.
Explanation

Consider a coil connected to a battery and a key. When key is pressed, current is increased from zero to maximum value. This varying current produces a changing magnetic flux around the coil. The coil is situated in this changing flux, so that an e.m.f. is produced in the coil.

This induced e.m.f. is produced in the coil. This induced e.m.f is opposite to applied e.m.f (E). Hence this induced e.m.f is called back e.m.f.

Similarly, when key is released, a back e.m.f is produced which opposes the decay of current in the circuit.

Thus both the growth and decay of currents in a circuit is opposed by the back e.m.f. This phenomenon is called self – induction.

Mathematical expression for self inductance :
Consider a solenoid (air core) of length /, number of turns N and area cross section A. let ‘n’ be the no. of turns per unit length (n = N/l)
The magnetic flux linked with the solenoid,
Φ = BAN
Φ = µ₀nIAN (since B = µ₀nI)
but Φ = LI
∴ LI = µ₀nIAN
L = µ₀nAN
If solenoid contains a core of relative permeability µ_r the L = µ₀µ_rnAN.

Definition of self inductance:
We know NΦ = LI
If I = 1, we get L = NΦ
Self inductance (or) coefficient of self induction may be defined as the flux linked with a coil, when a unit current is flowing through it.
Note: Physically, the self inductance plays the role of inertia.
Relation between induced emf and coefficient of self induction:
When the current through a coil is varied, a back emf produced in the coil. Using Lens law, emf can be written as,

4. Energy stored in an inductor:
When the current in the coil is switched on, a back emf (ε = -Ldt/dt) is produced. This back emf opposes the growth of current. Hence work should be done, against this e.m.f.
Let the current at any instant be ‘I’ and induced emf
E = \(\frac{-\mathrm{d} \phi}{\mathrm{dt}}\)
i.e., work done, dw= EIdt

Hence the total work done (when the current grows from 0 to I₀ is)

This work is stored as potential energy.
V = \(\frac{1}{2}\)LI²

Ac Generator
An ac generator works on electromagnetic induction. AC generator converts mechanical energy into electrical energy.

The structure of an ac generator is shown in the above figure. It consists of a coil. This coil is known as armature coil. This coil is placed in between magnets. As the coil rotates, the magnetic flux through the coil changes. Hence an e.m.f. is induced in the coil.

1. Expression for induced emf:

Take the area of coil as A and magnetic field produced by the magnet as B. Let the coil be rotating about an axis with an angular velocity ω.

Let θ be the angle made by the areal vector with the magnetic field B. The magnetic flux linked with the coil can be written as

Φ = BA cosθ
Φ = BA cosωt [since θ = ωt)
If there are N turns
Φ = NBA cosωt
∴ The induced e.m.f. in the coil,

Let ε₀ = NAB ω,
then s = ε₀ sin ωt.

Expression for current:
When this emf is applied to an external circuit .alternating current is produced. The current at any instant is given by
I = \(\frac{V}{R}\)

(V = ε₀ sin ω)
I = I₀ sin ωt
Where I₀ = ε₀/R, it gives maximum value of current. The direction of current is changed periodically and hence the current is called alternating current.

Variation of AC voltage with time:

Students can Download Chapter 9 Ray Optics and Optical Instruments Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 9 Ray Optics and Optical Instruments

Introduction
In this chapter, we consider the phenomena of reflection, refraction and dispersion of light, using the ray picture of light.

Reflection Of Light Byspherical Mirrors
Laws of reflection:

1. According to the first law of reflection, the angle of reflection equals the angle of incidence.
2. According to the second law of reflection, the incident ray, reflected ray and the normal to the point of incidence all lie in the same plane.

1. Sign convention:

• According to this convention, all distances are measured from the pole of the mirror or the optical centre of the lens.
• The distances measured in the same direction as the incident light are taken as positive and
  those measured in the direction opposite to the direction of incident light are taken as negative.
• The heights measured upwards are taken as positive. The heights measured downwards are taken as negative.

2. Focal length of spherical mirrors:
Reflection of light: Spherical mirrors are of two types.

• Concave mirror
• Convex mirror

Principal focus of a concave mirror:

A narrow parallel beam of light, parallel and close to the principal axis, after reflection converges to a fixed point on the principal axis is called principal focus of concave mirror.
Principal focus of a convex mirror:

A narrow parallel beam of light, parallel and close to the principal axis, after reflection appears to diverge from a point on the principal axis is called principal focus of convex mirror.
Relation connecting focal length and radius of curvature:

Consider a ray AB parallel to principal axis incident on a concave mirror at point B and is reflected along BF. The line CB is normal to the mirror as shown in the figure.
Let θ be angle of incidence and reflection.
Draw BD ⊥ CP,
In right angled ΔBCD,
Tanθ = \(\frac{B D}{C D}\) _____(1)
In right angled ΔBFD,
Tan2θ = \(\frac{B D}{F D}\) _____(2)
Dividing (1)and(2)
\(\frac{\tan 2 \theta}{\tan \theta}=\frac{C D}{F D}\) ____(3)
If θ is very small, then tanθ ≈ θ and tan2θ ≈ 2θ
The point B lies very close to P. Hence CD ≈ CP and FD ≈ FP From (3) we get

3. The mirror equation:

Let points P, F, C be pole, focus, and centre of curvature of a concave mirror. Object AB is placed on the principal axis. A ray from AB incident at E and then reflected through F. Another ray of light from B incident at pole P and then reflected. These two rays meet at M. The ray of light from point B is passed through C. Draw EN perpendicular to the principal axis.
ΔIMF and ΔENF are similar.
ie. \(\frac{I M}{N E}=\frac{I F}{N F}\) _____(1)
but IF = PI – PF and NF = PF (since aperture is small)
hence eq. (1) can be written as

[∵ NE = AB)
ΔABP and ΔIMP are similar

From eq.(2) and eq.(3), we get

applying sign convention we get
PI = -v
PF = -F
PA = -u
Substituting these values in eq.(4) we get

This is called mirror formula or mirror equation.
Linear magnification:
Linear magnification is defined as the ratio of the height of the image to the height of the object.

Consider an object AB having height h_o, which produces an image IM having height h_i
In the figure, ΔABP and ΔIMP are equal. ie.

Applying sign convention
PI = -V, PA = -u, h_i = -ve and h_o = +ve
We get

But we know \(\frac{h_{i}}{h_{0}}\) = m (magnification) ie.

This formulae is true fora concave mirror and convex mirror.
Relation connecting v, f, and m
We have
\(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\)
Multiplying throughout by ‘v’, we get
\(\frac{v}{u}+\frac{v}{v}=\frac{v}{f}\)
But m = -v/u
ie. -m + 1 = \(\frac{v}{f}\)
m = 1 – \(\frac{v}{f}\)

Relation connecting u, f and m
We know
\(\frac{1}{u}+\frac{1}{v}=\frac{1}{f}\)
Multiplying throughout by ‘u’ we get

Refraction
The phenomenon of bending of light when it travels from one medium to another is known as refraction.
Light from rarer to denser medium:

When light travels from a rarer medium to a denser medium, it deviates towards the normal.
Light from denser to rarer medium:

When light travels from a denser medium to a rarer medium, it deviates away from the normal.

Laws of refraction:
First law:
The incident ray, the refracted ray, and the normal at the point of incidence are all in the same plane.

Second law (Snell’s law):
The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant for a given pair of media and for the given colour of light used. This constant is known as the refractive index of second medium w.r. t. the first medium.

Explanation:
If ‘i’ is the angle of incidence in the first medium and ‘r’ is the angle of refraction in the second medium, then by Snell’s law,

Where ₁n₂ is the refractive index of the second medium with respect to the first medium. If the first medium is air, then sini/sinr is known as absolute refractive index of the second medium.
ie, \(\frac{\sin i}{\sin r}=n\)
where ‘n’ is the refractive index of the second medium.

Some examples of refraction:
(a) Apparent depth:

When an object (in a denser medium) is viewed from a rarer medium, it seems to be raised towards the surface. This is called apparent depth.

(b) Twinkling of stars:
Twinkling of stars is due to the refraction of star light at different layers of the atmosphere. Due to this refraction the star at S appears at S¹. But the density of the layer continuously changes. So, the apparent position continuously changes. Thus the star appears to be twinkling.

(c) Apparent shift in the position of the sun at sunrise and sunset:

Sun is visible before sunrise and after sunset because of atmospheric refraction. The density of atmospheric air decreases as we go up. So the rays coming from the sun deviates towards the normal. So the sun at ‘S’ appears to come from ‘S¹’. Thus an observer on earth can see the sun before sunrise and after sunset.

Total Internal Reflection

When a ray of light passes from a denser to rarer medium, after refraction the ray bends away from the normal. If the angle of incidence increases, the angle of refraction increases. When the angle of refraction is 90°, the corresponding angle of incidence is called the critical angle.

If we increases the angle of incidence beyond the critical angle, the ray is totally reflected back to the same medium. This phenomenon is called total internal reflection.

Relation between critical angle and refractive index
Refractive index,

where ‘C’ is the critical angle.
A demonstration for total internal reflection
Demonstration – 1:

Take a soap solution in a beaker. Now direct the laser beam from one side of the beaker such that it strikes the upper surface of water obliquely. Adjust the direction of laser beam until the beam is totally reflected back to water.

Demonstration – 2:

Take a soap solution in a long test tube and shine the laser light from top, as shown in above figure. Adjust the direction of the laser beam such that it is totally internally reflected. This is similar to what happens in optical fibres.
Condition for total internal reflection:

1. Light should travel from denser medium to rarer medium.
2. Angle of incidence in the denser medium should be greater than the critical angle.

Relative critical angle:
Critical angle of a medium A with respect to a rarer medium B is represented as _BC_A. _BC_A is related to the refractive index _Bn_A as

Some Effects And Applications Of Total Internal Reflection
(a) Brilliance of diamond:
Refractive index of diamond is high (n = 2.42) and the critical angle is small (C = 24.41°). More over the faces of the diamond are cut in such a way that a ray of light entering the crystal undergoes multiple total reflections. This multiple reflected light come out through one or two faces. So these faces appear glittering.

(b) Mirage:
On hot summer days the layer of air in contact with the sand becomes hot and rare. The upper layers are comparatively cooler and denser. When light rays travel from denser to rarer, they undergo total internal reflection. Thus image of the distant object is seen inverted. This phenomenon is Known as mirage.

(c) Looming (superior mirage):
Due to the mist and fog in cold countries, distant ship cannot be seen clearly. But due to the total internal reflection, the image of the ship appears hanging in air. This illusion is known as looming.

(d)Total reflection prisms:

A right-angled prism is called a total reflecting prisms. Total reflecting prisms are based on the principle of total internal reflection. With the help of these prisms, the direction of the incident ray can be changed. The refractive index for glass is 1.5 and its critical angle is 42°. When a ray of light makes an angle of incident more than 42° (within the glass) the ray undergoes total internal reflection.

1. Optical fibres:

Optical fibres consist of a number of long fibres made of glass or quartz (n = 1.7). They are coated with a layer of a material of lower refractive index (1.5). When light incident on the optical fibre at angle greater than the critical angle, it undergoes total internal reflection. Due to this total internal reflection, a ray of light can travel through a twisted path.
Uses:

• Used as a light pipe in medical and optical diagnosis.
• It can be used for optical signal transmissions.
• Used to carry telephone, television and computer signals as pulses of light.
• Used for the transmission and reception of electrical signals which are converted into light signals.

Refraction At Spherical Surfaces And By Lenses
Spherical lenses:
There are two types of lenses

• convex lenses and
• concave lenses.

Principal axis:
A straight line passing through the two centers of curvature is called the principal axis of the lens.

Principal focus (F):
A narrow beam of parallel rays, parallel and close to the principal axis, after refraction, converges to a point on the principal axis in the case of a convex lens or appears to diverge from a point on the axis in the case of a concave lens. This fixed point is called the principal focus of the lens.

Focal length:
It is distance between the optic centre and the principal focus.

1. Refraction at a spherical surface:

Consider a convex surface XY, which separates two media having refractive indices n₁ and n₂. Let C be the centre of curvature and P be the pole. Let an object is placed at ‘O’, at a distance ‘u’ from the pole. I is the real image of the object at a distance V from the surface. OA is the incident ray at angle ‘i’ and Al is the refracted ray at an angle ‘r’. OP is the ray incident normally. So it passes without any deviation. From snell’s law,

r₁ = n₂ _____(1)
From the Δ OAC, exterior angle = sum of the interior opposite angles
i.e., i = α + θ ______(2)
Similarly, from ΔIAC,
a = α + β
r = α – β ______(3)
Substituting the values of eq(2) and eq(3)in eqn.(1) we get,
n₁(α + θ) = n₂(α – β)
n₁α + n₁θ = n₂α – n₂β
n₁θ + n₂β = n₂α – n₁α
n₁θ + n₂β = (n₂ – n₁)α _______(4)
From OAP, we can write,

From IAP, β = \(\frac{\mathrm{AP}}{\mathrm{PI}}\), From CAP, α = \(\frac{\mathrm{AP}}{\mathrm{PC}}\)
Substituting θ, β and α in equation (4) we get,

According to New Cartesian sign convection, we can write,
OP = -u, PI = +v and PC = R
Substituting these values, we get

Case -1: If the first medium is air, n₁ = 1, and n₂ = n,

2. Refraction by a lens:
Lens Maker’s Formula (for a thin lens):
Consider a thin lens of refractive index n₂ formed by the spherical surfaces ABC and ADC. Let the lens is kept in a medium of refractive index n₁ Let an object ‘O’ is placed in the medium of refractive index n₁ Hence the incident ray OM is in the medium of refractive index n₁ and the refracted ray MN is in the medium of refractive index n₂.

The spherical surface ABC (radius of curvature R₁) forms the image at I₁. Let ‘u’ be the object distance and ‘v₁‘ be the image distance.
Then we can write,

This image I₁ will act as the virtual object for the surface ADC and forms the image at v.
Then we can write,

Adding eq (1) and eq (2) we get

Dividing throughout by n1, we get

if the lens is kept in air, \(\frac{\mathrm{n}_{2}}{\mathrm{n}_{1}}\) = n
So the above equation can be written as,

From the definition of the lens, we can take, when u = 8, f = v
Substituting these values in the eq (3), we get

This is lens maker’s formula

For convex lens,
f = +ve, R₁ = +ve, R₂ = – ve

For concave lens,
f = -ve, R₁ = -ve, R₂ = +ve

Lens formula

Linear magnification: If h_o is the height of the object and h_i is the height of the image, then linear magnification

3. Power of a lens:
Power of a lens is the reciprocal of focal length expressed in meter.

Unit of power is dioptre (D).

4. Combination of thin lenses in contact:

Consider two thin convex lenses of focal lengths f₁ and f₂ kept in contact. Let O be an object kept at a distance ‘u’ from the first lens L₁, I₁ is the image formed by the first lens at a distance v₁.
Then from the lens formula, we can write,

This image will act as the virtual object for the second lens and the final image is formed at I (at a distance v). Then

If the two lenses are replaced by a single lens of focal length ‘F’ the image is formed at V. Then we can write,

where P is the power of the combination, P₁ and P₂ are the powers of the individual lenses.

Magnification (combination of lenses):
If m₁, m₂, m₃,…….. are the magnification produced by each lens,
then the net magnification,
m = m₁. m₂. m₃……….
Relation connecting m, u and f:

Relation connecting m,v and f:

Refraction Through A Prism

ABC is a section of a prism. AB and AC are the refracting faces, BC is the base of the prism, ∠A is the angle of prism.
Aray PQ incidents on the face AB at an angle i₁. QR is the refracted ray inside the prism, which makes two angles r₁ and r₂ (inside the prism). RS is the emergent ray at angle i₂.
The angle between the emergent ray and incident ray is the deviation ‘d’.
In the quadrilateral AQMR,
∠Q + ∠R = 180°
[since and N₁M are normal] ie,
∠A + ∠M = 180° ____(1)
In the Δ QMR.
∴ r₁ + r₂ + ∠M = 180° _____(2)
Comparing eq (1) and eq (2)
r₁ + r₂ = ∠A ______(3)
From the Δ QRT,
(i₁ – r₁) + (i₂ – r₂) = d
[since exterior angle equal sum of the opposite interior angles]
(i₁ + i₂) – (r₁ + r₂) = d
but, r₁ + r₂ =A
∴ (i₁ + i₂ ) – A = d
(i₁ + i₂) = d + A _____(4)

It is found that for a particular angle of incidence, the deviation is found to be minimum value ‘D’.
At the minimum deviation position,
i₁ = i₂ = i, r₁ = r₂ = r and d = D
Hence eq (3) can be written as,
r + r= A
or r = \(\frac{A}{2}\) ______(5)
Similarly eq (4) can be written as,
i + i = A + D
i = \(\frac{A+D}{2}\) _____(6)
Let n be the refractive index of the prism, then we can write,
n = \(\frac{\sin i}{\sin r}\) ______(7)
Substituting eq (5) and eq (6) in eq (7),

i – d curve:

It is found that when the angle of incidence increases deviation (d) decreases and reaches a minimum value and then increases. This minimum value of the angle of deviation is called the angle of minimum deviation.

Dispersion By A Prism
Dispersion: The splitting of the white light into its component colours is called dispersion.

The pattern of colour components of light is called the spectrum of light.

Reason for dispersion:
The refractive index is different for different colours. Refractive index for violet is higher than red. This variation of refractive index of medium with the wavelength causes dispersion.

Some Natural Phenomena Due To Sunlight
1. The rainbow:
The rainbow is an example of the dispersion of sunlight by the water drops in the atmosphere. The conditions for observing a rainbow are that the sun should be shining in one part of the sky while it is raining in the opposite part of the sky.
There are two types rainbow

• Primary rainbow
• secondary rainbow.

(i) Primary rainbow:

In a primary rainbow, after refraction at the surface of water droplet, the ray suffers one internal reflection and finally comes out of the drop by forming an inverted spectrum. The maximum deviated light is red (42°) and the least deviated light is violet (40°).

(ii) Secondary rainbow:

secondary rainbow, after refraction at the surface of water droplet, the ray suffers two total internal reflection and finally comes out of the droplet by forming a spectrum. The most deviated light in this spectrum is violet (53°) and the least deviated light is red (50°).

2. Scattering of light:
When sunlight travels through the earth’s atmosphere, it changes its direction by atmospheric particles. This is called scattering. Light of shorter wavelength is scattered much more than light of longer wavelength. Scattering is possible only when size of the particles is comparable to the wavelength of incident light.

Rayleigh’s scattering law:
The intensity of the scattered light from a molecule is inversely proportional to the 4^th power of the wavelength.
ie, \(I \alpha \frac{1}{\lambda^{4}}\)
I – Intensity of Scattering

Blue colour of sky:
According to Rayleigh scattering, scattering is inversely proportional to the fourth power of its wavelength. Hence shorterwavelength is scattered much more than longer wavelength. Thus blue colour is more scattered than the other colours. So sky appears blue.

Whiteness of clouds:
Clouds contain large partides (dust, H₂O), which scatter all colours almost equally. Hence clouds appear white.

Colours of the sunset (or sunrise):
At sunrise and sunset light has to travel a longer distance before reaching the earth. During this time, smaller wavelengths are scattered away. The remaining colours is red. Hence sky appears red in colour.

Optical Instruments
Mirrors, lenses and prisms, periscope, Kaleidoscope, Binoculars, telescopes, microscopes are some examples of optical devices Our eye is one of the most important optical device.

1. The eye:

Human eye consists of an eyeball of size 2.5cm in diameter. The very thin skin in front of the eye is known as cornea. Behind cornea, the empty space is known as aqueous humor. The small wall behind cornea is known as iris. In this iris a small circular opening is there, which is known as pupil. Iris can adjust its tension to vary the size of the pupil.

Behind the iris a muscular membrane is there which is known as ciliary muscle. The focal length of the crystalline lens can be adjusted to see the object any separation by adjusting the tension of ciliary muscles. The backwall of eye is known as retina.

It consists of light sensitive cells known as rods and cones. The rods are sensitive to intensity and cones are sensitive to colour. The signals from retina are transferred to the brain by optic nerves.

The brightest point in the retina is known as yellow spot and the lowest point in the eye (retina) is known as blind spot. The space between the lens and retina is filled by a liquid which is known as vitreous humor.

Defects of Vision
a. Myopia or shortsightedness:

A person suffering from myopia can see only nearby objects but cannot see objects beyond a certain distance clearly. This defect occurs due to

1. Elongation of eyeball
2. Short focal length of eye lens

It can be corrected by using a concave lens of suitable focal length.

b. Hypermetropia or Far sightedness:

A person suffering from this defect can see only distant object clearly but cannot see nearby objects clearly. This defect occurs due to

1. Decrease in the size of the eyeball.
2. Increase in focal length of the eyeball.

This defect can be corrected by using a converging lens (convex).

c. Astigmatism:

A person suffering from astigmatism cannot focus objects in front of the eye clearly. It can be corrected by using a cylindrical lens of suitable focal length.

d. Presbyopia:
It is the farsightedness occurring due to awakening of ciliary muscles. It can be corrected by using a lens of bifocal length.

1. The microscope:
Simple microscope: A simple microscope is a converging lens of small focal length.

Working: The object to be magnified is placed very close to the lens and the eye is positioned close to the lens on the other side. Depending upon the position of object, the position of image is changed.

Case 1:
If the object is placed, one focal length away or less, we get an erect, magnified and virtual image at a distance so that it can be viewed comfortably ie. at 25cm or more. (This 25cm is denoted by the symbol D).

Case 2:
If the object is placed at a distance f (focal length of lens), we get the image at infinity.

Mathematical expression of magnification:
Image at D:
If the image is formed at ‘D’, we can take u = -D. Hence the lens formula can be written as

The image is formed at D, ie. v = -D

This equation is used to find magnification of simple microscope when image at D (D ≈ 25cm).

Image at infinity:

If the object is placed at f, the image forms at infinity. In this case, magnification,

Suppose the object has a height h, the angle subtended is
tanθ₀\(=\frac{h}{D}\), θ₀\(=\frac{h}{D}\)______(2)
where ‘D’ is the comfortable distance of object from the eye (least distinct vision).

When the final image is formed at infinity,
θ_i = \(\frac{h^{1}}{v}\) ______(3)
When h₁ is the height of image and v is the image distance

This equation is used to find magnification of simple microscope when image at infinity.

2. Compound microscope:
Apparatus: A compound microscope consists of two convex lenses, one is called the objective and the other is called eye piece.

The convex lens near to the object is called objective. The lens near to the eye is called eye piece. The two lenses are fixed at the ends of two co-axial tubes. The distance between the tubes can be adjusted.

Working:
The object is placed in between F and 2F of objective lens. The objective lens forms real inverted and magnified image (I₁M₁) on the other side of the lens.

This image will act as object or eyepiece. Thus an enlarged, virtual, and inverted image is formed, (this image can be adjusted to be at the least distance of distinct vision, D).

Magnification: The magnification produced by the compound microscope

Where m₀ & m_e are the magnifying power of objective lens and eyepiece lens.

Eyepiece acts as a simple microscope.
Therefore m_e = 1 + \(\frac{D}{f_{e}}\) _____(2)
m₀ = \(\frac{v_{0}}{u_{0}}\) ______(3)
We know magnification of objective lens
Where v₀ and u₀ are the distance of the image and object from the objective lens.
Substituting (2) and (3) in (1), we get

for compound microscope, u_o » f_o (because the object of is placed very close to the principal focus of the objective) and v_o ≈ L, length of microscope (because the first image is formed very close to the eye piece).

where L is the length of microscope, f₀ is the focal length of objective lens.
Case 1: If the final image is formed at infinity, magnification of eye piece D
m \(=\frac{D}{f_{e}}\)
∴ Total magnification of compound microscope

(3) Telescope: Astronomical telescope is used to observe heavenly bodies.
There are two types of telescopes

1. Refracting and
2. Reflecting Telescope.

(1) Refracting Telescope:
Constructional details

It consists of two convex lenses, one is called objective and other is called eyepiece. These two lenses are fitted at the ends of two coaxial tubes. The distance between the two lenses can be varied.

Working:
The objective lens forms the image (IM) of a distant object at its focus. This image (formed by objective) is adjusted to be focus of the eyepiece.

Magnification:
The magnifying power of a telescope is the ratio of the angle subtended by the image at the eye to the angle subtended by the object at the objective.

(For small values tan α ≈ α)

But IC = f_o (the focal length objective lens) and IC¹ = f_e(the focal length eyepiece lens.)
∴ m = \(\frac{f_{0}}{f_{e}}\)
In this case the length of the telescope tube is (f₀ + f_e).

Case 1: When the image formed by the objective is within the focal length of the eyepiece, Then the final image is formed at the least distant of distinct vision. In this case, magnifying power.

(2) Reflecting Telescope:
Newtonian types reflecting Telescope:
The Newtonian reflector consists of a parabolic mirror made of an alloy of copper and tin. It is fixed atone end of a metal tube.

The parallel rays from a distant stars incident on the mirror M₁. After reflection from the mirror, the ray incident on a plane mirror M₂.

The reflected ray from M₂ enter into eye piece E. The eyepiece forms a magnified, virtual and erect image. Magnifying power of Newton Telescope
m = \(\frac{f_{0}}{f_{e}}\) or m = \(\frac{R}{2 f_{\theta_{g}}}\)
where
f_o — is the focal length of concave mirror
f₂ — is the focal length of eyepiece.
R – Radius of curvature of concave reflector.

Students can Download Chapter 8 Electromagnetic Waves Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 8 Electromagnetic Waves

Introduction
In this chapter we shall study the basic concepts of electromagnetic waves.

Displacement Current
Amperes circuital law in ac circuit: Consider a capacitor connected to a AC source using conducting wires. AC current can flow through a capacitor. Hence magnetic field is produced around the conducting wire. This magnetic field can be found using amperes circuital law.

Magnetic field at P
Method – 1 (To find magnetic field at P)

Consider a point P, which lies outside and very close to a capacitor as shown in the figure. We can find magnetic field at P using amperes circuital law. In order to find an magnetic field at P, consider a open surface (amperien loop having pot like surface) with a boundary of circle of radius r.
Applying amperes circuital law we get
\(\oint\)B.dI = µ₀i
Where ‘i’ is the current passing through the surface. (This surface lies outside to capacitor)
Integrating we get B.2πr = µ₀i
B = \(\frac{\mu_{0} i}{2 \pi r}\) _____(1)

Method – 2 (To find magnetic field at P)

Consider a open surface (amperien loop having pot like surface) extended to interior of capacitor with a boundary of circle of radius r.
Applying amperes circuital law we get
\(\oint\)B.dI = µ₀0
(since the current passing through the closed surface is zero, surface lies in between the plates)
ie. B = 0 ______(2)

Discussion of method 1 and method 2: Amperian circuital law is independent of size and shape of pot like surface. Hence we expect same value of B in eq(1)and eq(2). But we got different values at the same point P. Hence we can understand that there is a mistake in the amperes circuital law in AC circuits.

Maxwells correction in amperes circuital law:
To solve the above mistake, Maxwell introduced a term in the amperes circuital law. The modified amperes circuital law can be written as
\(\oint\)B.dI = µ₀(i_c + i_d)
Where i_d is called displacement current. Its value is

The above modified amperes circuital is known as Ampere- Maxwell law. This law is applicable for both AC and DC circuits.

Question 1.
Show that conduction current i_c is equal to displacement current i_d
Answer:
The flux passing through the surface in between plates (see figure 2)

Capacitor is connected to ac voltage. Hence the charge on the plate also changes with time. Hence the flux passing through the pot shape surface changes with time.
ie. the flux in between capacitor changes.
The change influx,

This means that the conduction current passing through the conduction wire is converted into displacement current, when it passes in between plates of capacitor.
1. The total current i is the sum of the conduction current and the displacement current
So we have

2. Outside the capacitor plates, we have only conduction current and no displacement current inside the capacitor there is no conduction current and there is only displacement current.

Electromagnetic Waves
1. Sources of Electromagnetic waves:
Question 2.
How are electromagnetic waves produced?
Answer:
Consider a charge oscillating with some frequency (An oscillating charge is an example of accelerating charge). This oscillation produces an oscillating electric and magnetic field in space. The oscillating electric and magnetic fields (EM Wave) propagates through the space. The experimental production of electromagnetic wave was done by Hertz’s experiment in 1887.

2. Nature of electromagnetic waves:
Characteristics of Electromagnetic waves:
(i) Electromagnetic waves propagate in the form of mutually perpendicular magnetic and electric
fields. The direction of propagation of wave is perpendicular to both magnetic and electric field vector.

(ii) Velocity of electromagnetic waves in free space is,

The speed of electromagnetic wave in a material medium is given by

(iii) The ratio of magnitudes of electric and magnetic field vectors in free space is constant

E and B are in same phase

(iv) No medium is required for propagation of transverse wave.

(v) Electromagnetic waves show properties of reflection, refraction, interference, diffraction and polarization.

(vi) Electromagnetic waves have capability to carry energy from one place to another.

Mathematical Expression:

Consider a plane electromagnetic wave travelling along the Z direction. The electric and magnetic fields are perpendicular to the direction of wave motion.
The electric field vector along the Y direction.
E_x = E₀sin(kz – ωt)
and B_Y = B₀sin(kz – ωt)
where E₀ is the amplitude of electric field vector, B₀ is the amplitude of magnetic field vector, ω is the angular frequency and k is related to the wave length λ of the wave,
k = 2π/λ.

Electromagnetic Spectrum
Electromagnetic waves include visible light waves, X-rays, gamma rays, radio waves, microwaves, ultraviolet and infrared waves. The classification is based roughly on how the waves are produced or detected.

1. Radio waves:
Radio waves are produced by the accelerated motion of charges in conducting wires. They are used in radio and television communication systems. They are generally in the frequency range from 500 kHz to about 1000 MHz.

2. Microwaves:
Microwaves (short-wavelength radio waves), with frequencies in the gigahertz (GHz) range, are produced by special vacuum tubes (called klystrons, magnetrons, and Gunn diodes). Due to their short wavelengths, they are suitable for the radar systems used in aircraft navigation. Microwave ovens are domestic application of these waves.

3. Infrared waves:
Infrared waves are produced by hot bodies and molecules. Infrared waves are sometimes referred to as heatwaves. Infrared lamps are used in physical therapy.

Infrared rays are widely used in the remote switches of household electronic systems such as TV, video recorders etc. Infrared radiation also plays an important role in maintaining the earth’s warmth or average temperature through the greenhouse effect.

4. Visible rays:
It is the part of the spectrum that is detected by the human eye. It starts from 4 × 10¹⁴ Hz to 7 × 10¹⁴ Hz (ora wavelength range of about 700 – 400 nm).

5. Ultraviolet rays (UV):
It covers wavelengths ranging from about 4 × 10^-7m to 6 × 10^-10m (0.6 nm to 400 nm)). UV radiation is produced by special lamps and very hot bodies. The sun is an important source of ultraviolet light.

UV light in large quantities has harmful effects on humans. Exposure to UV radiation induces the production of more melanin, causing tanning of the skin. UV radiation is absorbed by ordinary glass. Hence, one cannot get tanned or sunburn through glass windows.

Due to its shorter wavelengths, UV radiations can be focussed into very narrow beams for high precision applications such as eye surgery. UV lamps are used to kill germs in water purifiers.

6. X-rays:
It covers wavelengths from about 10^-8m to 10^-13m (4nm – 10nm). One common way to generate X-rays is to bombard a metal target by high energy electrons. X-rays are used as a diagnostic tool in medicine and as a treatment for certain forms of cancer.

7. Gamma rays:
They lie in the upper-frequency range of the electromagnetic spectrum and have wavelengths of^rom about 10^-10m to less than 10^-14m. This high-frequency radiation is produced in nuclear reactions and also emitted by radioactive nuclei. They are used in medicine to destroy cancer cells.

Different Types Of Electromagnetic Waves:

Students can Download Chapter 7 Alternating Current Notes, Plus Two Physics Notes helps you to revise the complete Kerala State Syllabus and score more marks in your examinations.

Kerala Plus Two Physics Notes Chapter 7 Alternating Current

Alternating Current
AC current is commonly used in homes and offices. The main reason for preferring ac voltage over dc voltage is that ac voltages can be easily converted from one voltage to the other and can be transmitted over long distances. In this chapterwe will deal the properties of ac and its flowthrough different devices (inductor, capacitor, etc).

Ac Voltage Applied To A Resistor

Consider a circuit containing a resistance ‘R’ connected to an alternating voltage.
Let the applied voltage be
V = V_o sinωt ______(1)
According to Ohm’s law, the current at any instant can be written as
I = \(\frac{V_{0} \sin \omega t}{R}\)
Where I₀ = V_o/R is the peak value of current. Comparing eq(1) and eq(2), we can understand that the current and voltage are in same phase.
Graphical variation of current and voltage:

R.M.S value (or Virtual value, effective value) of current and voltage:
The mean value of emf and current for one cycle is zero. Hence to measure ac, the root mean square (rms) values are considered.

The r.m.s value or virtual value of an AC is the square root of the mean of the squares of the instantaneous value of current taken over a complete cycle.
I_rms = \(\frac{I_{0}}{\sqrt{2}}\) and V_rms = \(\frac{V_{0}}{\sqrt{2}}\)
where I₀ – maximum current, V₀ – maximum voltage, (r.m.s.- root mean square).

Power dissipated in the resistor:
The average power consumed in one complete cycle,

Substituting current and voltage, We get

Representation Of Ac Current And Voltage By Rotating Vectors – Phasors
To represent the phase relation between current and voltage, phasors are used. Aphasoris a vector which rotates about the origin with an angular speed ω. The vertical components of phasors of V and I represent instantaneous value of V and I at a time t (see figure). The length of phasors give maximum amplitudes of V and I.

Phasor diagram of v and i for the circuit containing resistor only

The figure(a) represent the voltage and current phasors and their relationship at time t₁. Fig (b) shows the graphical variation of V and I.

Ac Voltage Applied To An Inductor

Consider a circuit containing an inductor of inductance ‘L’ connected to an alternating voltage.
Let the applied voltage be
V = V_o sinωt _____(1)
Due to the flow of alternating current through coil, an emf, \(\frac{d I}{d t}\) is produced in the coil. This induced emf is equal and opposite to the applied emf (in the case of ideal inductor)

Integrating, we get

Where I_o = \(\frac{V_{0}}{L \omega}\),
The term Lω is called inductive reactance. Comparing eq(1) and eq(2), we can understand that, the current lags behind the voltage by an angle 90°.
Graphical variation of current and voltage:

Phasor diagram:

Inductive reactance X_L:
The resistance offered by an inductor to a.c. flow is called inductive reactance.
Inductive reactance

Power Consumed by an Inductor Carrying AC:
The instantaneous value of voltage and current in a pure inductor is
V = V_o sinωt
I = I_o cosωt
The average power consumed per cycle.

The above expression indicates that the average power or net energy consumed by an inductor carrying ac for a full cycle is zero.

Ac Voltage Applied To A Capacitor

Consider a circuit containing a capacitor of capacitance ‘C’ connected to alternating voltage.
Let the applied voltage be V = V_o sinωt _____(1)
The instantaneous current through capacitor

Substituting eq.(1) in eq.(2), we get

\(\frac{1}{\mathrm{C} \omega}\) is called capacitative reactance
Comparing eq(1) and eq(3), we can understand that, the current leads the voltage by an angle 90°
Graphical variation of current and voltage:

Phaser diagram:

Capacitative Reactance X_c:
The resistance offered by a capacitor to ac flow is called Capacitative reactance
Capacitative reactance

1. Power consumed by a capacitor carrying current:
The instantaneous value of voltage and current in a pure inductor is
V = V_o sinωt
I = I_o cosωt
The average power consumed per cycle.

The above expression indicates that the average power or net energy consumed by a capacitor carrying ac for a full cycle is zero.

Ac Voltage Applied To A Series Lcr Circuit

Consider a circuit containing an inductance L, resistance R and capacitance C connected in series. An alternating voltage V = V_o sinωt is applied to the circuit.
Phasor Diagram:

Let V_R be the voltage across R. This voltage is represented by a vector OA (since I and V_R are in same direction). Let V_L be the voltage across L. This voltage is represented by a vector OB (since the voltage V_L leads the current by angle 90°).

Similarly, let V_c be the voltage across C. This voltage is represented by a vector OC (since the voltage V_c lags the current by angle 90°).

The phase difference between V_L and V_c is Φ(ie. they are in opposite directions). So the magnitude of net voltage across the reactance is (V_L – V_c). This is represented by a vector OD in phasor diagram.

The final voltage in the circuit is the vector sum of V_R and (V_L – V_c). The final voltage is represented by diagonal OE.

1. Impedances of LCR circuit:
From the right angled triangle OAE,
Final voltage, V = \(\sqrt{\mathrm{V}_{\mathrm{R}}^{2}+\left(\mathrm{V}_{\mathrm{L}}-\mathrm{V}_{\mathrm{c}}\right)^{2}}\)

Where Z is called impedance of LCR circuit

Phase Difference: Let Φ be the phase difference between final voltage V and current I
From fig (2), we can write

Expression for current:
The eq(2) shows that there is a phase difference between current and voltage. The instantaneous current lags the voltage by an angle (Φ).
If V = V_o sinωt is the applied voltage, the current at any instant can be written as
I = I_o sin(ωt – Φ) _____(3)
Where I_o is the peak value of current. It’s value can be written as

2. Analytical solution:
If we apply V = V_msinωt to an LCR circuit, we can write
V_L + V_R + V_C = V_m sinωt

Substituting these values in eq.(1), we get

The above equation (2) is like the equation for a forced, damped oscillator. Hence we can take the solution of above equation as
q = q_m sin(ωt + θ)

Substituting these values in eq.(2) we get

Multiplying and dividing by Z = \(\sqrt{R^{2}+\left(X_{0}-X_{L}\right)^{2}}\), we have

Substituting these values in eq.(4), we get
q_mωz[cosΦcos(ωt + θ) + sinΦsin(ωt + θ)] = V_msinωt
q_mωz cos(ωt + θ – Φ) = V_m sinωt ______(5)
(∴ cos(A – B) = cosA cosB + sinA sinB)
Comparing the two sides of the eq.(5) we get
V_m = q_mωz = i_mz
where i_m = q_mω
and cos(ωt + θ – Φ) = sinωt
sin (ωt + θ – Φ + π/2) = sinωt (∵ sin(θ + π/2) = cosθ)
ωt + θ – Φ + π/2 = ωt
(θ – Φ) = -π/2
Therefore, the current in the circuit is

Thus, the analytical solution for the amplitude and phase of the current in the circuit agrees with that obtained by the technique of phasors.

3. Resonance:
When ωL = \(\frac{1}{\omega C}\), the impedance of the LCR circuit becomes minimum. Hence current becomes maximum. This phenomena is called resonance.

The frequency of the applied signal at which the impedance of LCR circuit is minimum and current becomes maximum is called resonance frequency.
Expression for resonance frequency
Resonance occurs at ωL = \(\frac{1}{\omega C}\)

Impedance at resonance: Resonance occurs at ωL = \(\frac{1}{\omega C}\). Substituting this condition in eq(1), in section 7.6, we get
Impedance, Z = R
Current at resonance:
substituting ωL = \(\frac{1}{\omega C}\) in eq(2) in section 7.6 we get,
TanΦ = 0, or Φ = 0
Substituting this value in eq(3) in section 7.6, we get
current I= I_o sin ωt
Where I_o = V_o/R
Graphical variation of current with ω in LCR circuit:

Variation of current through LCR circuit with angular frequency co for two cases (1) R= 100Ω and (2) R=200Ω, is shown in the graph
Note: It is important to note that resonance phenomenon is exhibited by a circuit only if both L and C are present in the circuit. Only then the voltages across L and C cancel each other (both being out of phase).

The current amplitude ( V_m/R) is the total source voltage appearing across R. This means that we cannot have resonance in a RL or RC circuit.

4. Sharpness of resonance:

The figure shows the variation of current i with © in a LCR circuit.
Bandwidth: At ω₀, the current in the LCR circuit is maximum. Suppose we choose a value of ω for which the current amplitude is \(\frac{1}{\sqrt{2}}\) times its maximum value.

We can see that there are two values of ω(ω₁ and ω₂) and forwhich current is \(\frac{i_{m}}{\sqrt{2}}\).

The difference ω₂ – ω₁ is called bandwidth.
If we take ω₁ = ω₀ – ∆ω and ω₂ = ω₀ + ∆ω
We get bandwidth, ω₂ – ω₁ = 2∆ω.

Expression for bandwidth and sharpness of resonance:
We know that the current in the LCR circuit

We know that the current in the LCR circuit becomes \(\frac{i_{m}}{\sqrt{2}}\) at ω₂ = ω₀ + ∆ω. Substituting this m in eq(1), we get

But ω₂ = ω₀ + ∆ω, substituting this above equation.

Sharpness of resonance: The quantity \(\left(\frac{\omega_{0}}{2 \Delta \omega}\right)\) is
called sharpness of resonance.
From eq.(4),weget

When bandwidth increases, the sharpness of resonance decreases, ie. the tuning of the circuit will not be good.

Quality Factor (Q): The ratio \(\frac{\omega_{0} L}{R}\) is called the quality factor. When R is low or L is large, the quality factor becomes large. Lange quality factor means that the circuit is more selective.

Power in AC circuit: The power factor
Power in AC circuit with LC and R: In ac circuit the Voltage vary continuously.
∴ The average power in the circuit for one full cycle of period,


(since sin 2A = 2sinA CosA)
The mean value of sin²ωt over a complete cycle is 1/2 and the mean value of sin2ωt over a complete cycle is zero.

True power = Apparent power × power factor
The term P_av called true power. V_rms × I_rms is called the apparent power and cosΦ is called power factor.
powerfactor = \(\frac{\text { True power }}{\text { apparent power }}\)
Power factor is defined as the ratio of true power to apparent power.

Case – 1 (In purely resistive circuit)
In this case, current and voltage are in same phase. Hence Φ = 0
∴ P_av = V_rms I_rmsCosO
True power, P_av = V_rms I_rms

Case – 2 (In a purely inductive and purely capacitative circuit (no resistance)). In this case, the angle between voltage and current is 90°.
∴ P_av = V_rms I_rmsCos 90
True power, P_av = 0
Which means that, the power consumed by the circuit is zero. The current in such a circuit (purely inductive and purely capacitive) doesn’t do any work. A current that does not do any work is called wattles or idle current.

Lc Oscillations

A capacitor can store electrical energy. An inductor can store magnetic energy. When a charged capacitor is connected to an inductor, the electrical energy( of capacitor) transfers to magnetic energy (of inductor) and vise versa. Thus energy oscillates back and forth between capacitor and inductor. This is called L. C. Oscillations.
Expression for frequency:
Applying Kirchoff’s second rule, we get

Transformers
Principle: It works on the principle of mutual induction.
Construction:

A transformer consists of two insulated coils wound over a core. The coil, to which energy is given is called primary and that from which energy is taken is called secondary.

Working and mathematical expression :
Let V₁ N₁ be the voltage and number of turns in the primary. Similarly, let V₂, N₂ be voltage and number of turns in the secondary.

When AC is passed, a change in magnetic flux is produced in the primary. This magnetic flux passes through secondary coil.

If Φ₁ and Φ₂ are the magnetic flux of primary and secondary, we can write Φ₁ α N₁ and Φ₂ α N₂.
Dividing Φ₁ and Φ₂
\(\frac{\phi_{1}}{\phi_{2}}=\frac{N_{1}}{N_{2}}\)
[since Φ is proportional to number of turns] or \(\phi_{1}=\frac{\mathrm{N}_{1}}{\mathrm{N}_{2}} \phi_{2}\)
Taking differentiation on both sides we get

Step up Transformer:
If the output voltage is greater than input voltage, the transformer is called step up transformer. In a step up transformer N₂ > N₁ and V₂ > V₁.

Step down transformer:
If the output voltage is less than the input voltage, then the transformer is called step down transformer. In a step down transformer N₂ < N₁ and V₂ < V₁.

Efficiency of a transformer:
The efficiency of a transformer is defined as the ratio of output power to input power.

For an ideal transformer, efficiency = 1
i.e, V₁I₁ = V₂I₂

1. Power losses in a transformer
(i) Joule loss or Copper loss:
When current passes through a coil heat is produced. This energy loss is called Joule loss. It can be minimized by using thick wires.

(ii) Eddy current loss: This can be minimized by using laminated cores. Laminated core increases the resistance of the coil. Thus eddy current decreases.

(iii) Hysteresis loss: When the iron core undergoes cycles of magnetization, energy is lost. This loss is called hysteresis loss. This is minimized by using soft iron core.

(iv) Magnetic flux loss:
The total flux linked with the coil may not pass through secondary coil. This loss is called magnetic flux loss. This loss can be minimized by closely winding the wires.