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Electricity and Magnetism
Magnetic Induction
Magnetic induction refers to the production of an electromotive force (EMF) or current in a conductor when it is exposed to a changing magnetic field or when relative motion exists between the conductor and the magnetic field. It is governed fundamentally by Faraday's laws of electromagnetic induction and Lenz's law.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Faraday's laws of electromagnetic induction, Lenz's law, magnitude and direction of induced EMF, self-induction, mutual induction, and simple numericals on magnetic flux linked with coils.
PGT extension
Differential and integral forms of Maxwell's equations relating to electromagnetic induction, vector potential, eddy currents, betatron principle, and complex mutual inductance problems in coupled circuits.
Prerequisites
Biot-Savart Law and Ampere's Circuital Law
Magnetic flux and its spatial variation
Basic vector calculus and divergence/curl operations (for PGT)
Faraday's Laws of Electromagnetic Induction
First Law: Whenever the magnetic flux linked with a closed circuit changes, an induced electromotive force (EMF) is set up in the circuit.
Second Law: The magnitude of the induced EMF is directly proportional to the time rate of change of magnetic flux linked with the circuit.
Mathematical expression: $e = -dPhi/dt$ for a single turn, and $e = -N(dPhi/dt)$ for a coil of N turns.
Lenz's Law and Conservation of Energy
Lenz's Law states that the polarity of the induced EMF is such that it tends to oppose the very cause (change in magnetic flux) that produces it.
The negative sign in Faraday's law equation explicitly incorporates Lenz's law.
Lenz's law is a direct consequence of the law of conservation of energy; mechanical work done in moving a conductor through a magnetic field transforms into electrical energy.
Self-Induction and Self-Inductance
Self-induction is the property of a single coil by virtue of which it opposes any change in the strength of current flowing through it by inducing an opposing EMF.
The magnetic flux linked with the coil is directly proportional to the current flowing through it: Ph$i = L * I$.
Self-inductance (L) depends on the geometry of the coil, number of turns, and the permeability of the core medium, with SI unit Henry (H).
Mutual Induction and Mutual Inductance
Mutual induction is the phenomenon where a changing current in one coil induces an EMF in a neighboring secondary coil.
The flux linked with the secondary coil is proportional to the current in the primary coil: Ph$i_2 = M * I_1$.
Mutual inductance (M) depends on the separation, relative orientation, and magnetic coupling between the two coils.
Derivations
Self-Inductance of a Long Solenoid
Conditions: Long air-cored solenoid of length l, cross-sectional area A, and total turns N.
Write the magnetic field inside the long solenoid: $B = \mu_0 * n * I$, where $n = N / l$ is the number of turns per unit length.
Calculate the magnetic flux linked with a single turn: Ph$i_{single} = B * A = \mu_0 * (N/l) * I * A$.
Calculate the total magnetic flux linked with all N turns: Ph$i_{total} = N * \phi_{single} = \mu_0 * (N^2 / l) * A * I$.
Equate total flux to L * I to find the self-inductance: L * $I = \mu_0 * (N^2 / l) * A * I$.
Result: $L = \mu_0 * N^2 * A / l$
Induced EMF in a Rod Rotating in a Uniform Magnetic Field
Conditions: A metallic rod of length l rotating with angular velocity omega in a uniform magnetic field B perpendicular to the plane of rotation.
Consider an infinitesimal element of the rod of length dr at a distance r from the axis of rotation.
The linear velocity of this element is $v = \omega * r$.
The motional EMF induced across the element dr is d$e = B * v * dr = B * \omega * r * dr$.
Integrate the infinitesimal EMF from r = 0 to r = l to find the total induced EMF: $e = \int_{0}^{l} B \omega r $, dr.
Result: $e = \frac{1}{2} B \omega l^2$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Faraday's Law of Induction
$$e = -N \frac{d\Phi}{dt}$$
e = Induced EMF, N = Number of turns, d\Phi/dt = Rate of change of magnetic flux Volt (V)
Valid for any closed circuit in a changing magnetic field
Motional EMF
$$e = Blv$$
B = Magnetic field, l = Length of conductor, v = Velocity perpendicular to B Volt (V)
Conductor moving perpendicular to uniform magnetic field
Self-Inductance Definition
$$e = -L \frac{dI}{dt}$$
e = Back EMF, $L = Self-inductance$, dI/dt = Rate of change of current Henry (H)
Single isolated coil or circuit
Self-Inductance of Solenoid
$$L = \frac{\mu_0 \mu_r N^2 A}{l}$$
N = Total turns, A = Area, l = Length, \m$u_r = Relative permeability$ Henry (H)
Long solenoid approximation
Mutual Inductance Between Coils
$$M = \frac{\Phi_2}{I_1} = \frac{e_2}{-dI_1/dt}$$
M = Mutual inductance, \Ph$i_2 = Flux in coil 2$, $I_1 = Current in coil 1$ Henry (H)
Coupled magnetic circuit
Energy Stored in an Inductor
$$U = \frac{1}{2} L I^2$$
U = Magnetic potential energy, L = Inductance, I = Current Joule (J)
Steady current I established in inductor L
Magnetic Energy Density
$$u = \frac{B^2}{2 \mu_0}$$
u = Energy per unit volume, B = Magnetic field Joule per cubic meter (J/m^3)
Uniform magnetic field in free space
Coefficient of Coupling
$$k = \frac{M}{\sqrt{L_1 L_2}}$$
k = Coupling coefficient, M = Mutual inductance, L_1, $L_2 = Self-inductances$ Dimensionless
0 <= k <= 1
Graphs and experiments
Magnetic Flux vs. Time Graph
Draw: A curve plotting magnetic flux (Phi) on the y-axis against time (t) on the x-axis.
Interpret: The instantaneous slope (negative derivative) at any point on the curve gives the magnitude of the induced EMF at that exact instant.
Exam use: Used in conceptual multiple-choice questions to determine times of maximum and zero induced EMF.
Growth of Current in an L-R Circuit
Draw: An exponential growth curve of current (I) against time (t) asymptoting to maximum value $I_0 = E/R$.
Interpret: The time constant ($\tau = L/R) represents the time required$ for the current to reach approximately 63.2% of its maximum steady-state value.
Exam use: Tested numerically and conceptually for transient behavior in inductive circuits.
Electromagnetic Induction Experiment (Magnet and Coil)
Draw: A bar magnet moving toward a solenoid connected to a sensitive center-zero galvanometer.
Interpret: Demonstrates that relative motion generates current; reversing the motion or polarity of the magnet reverses the galvanometer deflection direction.
Exam use: Asked in descriptive or foundational principle questions regarding Faraday and Lenz laws.
Solved numericals
TGT core
A coil of 500 turns has a magnetic flux of 0.02 Wb linked with it. If the flux is reduced to zero in 0.1 seconds, find the induced EMF in the coil.
Identify given parameters: N = 500, initial flux Ph$i_1 = 0.02 Wb$, final flux Ph$i_2 = 0$, time interval Delta t = 0.1 s.
Two coaxial solenoids each of length 1 m have inner radius 2 cm and outer radius 3 cm. The inner solenoid has 1000 turns and the outer solenoid has 1500 turns. Find the mutual inductance between them.
Identify overlapping cross-sectional area common to both solenoids, determined by the inner radius $r = 0.02 m: A = \pi * r^2 = \pi * (0.02)^2 = 4pi * 10^{-4} m^2$.
Write the formula for mutual inductance of coaxial solenoids: $M = \mu_0 * (N_1 * N_2 / l) * A$.
Substitute the given values: m$u_0 = 4pi * 10^{-7}$, $N_1 = 1000$, $N_2 = 1500$, l = 1 m, $A = 4pi * 10^{-4} m^2$.
Solenoid formulas assume an infinitely long solenoid where edge effects and magnetic flux leakage at the ends are negligible.
Uniformity of magnetic field is assumed over the entire cross-sectional area of the loop unless spatial integration is explicitly performed.
Common misconceptions
Common error: An EMF is induced only when the magnetic field B itself changes.
Correct: An EMF is induced whenever the total magnetic flux (Ph$i = B * A * \cos\theta) changes$, which can happen by changing B, area A, or angle \theta.
Students often confuse magnetic field strength with magnetic flux linkage.
Common error: Lenz's law implies that the induced EMF cancels out the applied change completely.
Correct: Lenz's law opposes the change, but it cannot prevent the change from happening; otherwise, continuous induction would violate energy conservation.
Misinterpretation of the word 'opposition' leads to the belief that net flux cannot change.
Quick revision
Induced EMF depends strictly on the rate of change of magnetic flux, not on the total flux value itself.
Faraday's Law of Induction: $e = -N \frac{d\Phi}{dt}$
Motional EMF: e = Blv
Must remember
Faraday's second law states that the magnitude of induced EMF is $e = -N \frac{d\Phi}{dt}$.
Lenz's law is a direct consequence of the law of conservation of energy.
Self-inductance of a long solenoid is given by $L = \frac{\mu_0 \mu_r N^2 A}{l}$.
Magnetic potential energy stored in an inductor carrying current $I$ is $U = \frac{1}{2} L I^2$.
Magnetic energy density in free space is $u = \frac{B^2}{2 \mu_0}$.
Coefficient of coupling between two coils is defined as $k = \frac{M}{\sqrt{L_1 L_2}}$, where $0 \le k \le 1$.
Glossary
Magnetic Flux ($\Phi$)The measure of the total magnetic field passing through a given surface, expressed as $\Phi = \int \vec{B} \cdot d\vec{A}$.
Self-InductionThe property of a coil by which it opposes any change in the strength of current flowing through it by inducing a back EMF.
Mutual InductionThe phenomenon where a changing current in a primary coil induces an EMF in a neighboring secondary coil.
Motional EMFThe EMF induced across a conductor moving across a magnetic field, given by $e = Blv$.
Self-Induction vs Mutual Induction
Feature
Self-Induction
Mutual Induction
Number of Coils
Single coil
Two or more coupled coils
Governing Equation
$e = -L \frac{dI}{dt}$
$e_2 = -M \frac{dI_1}{dt}$
Unit of Inductance
Henry (H)
Henry (H)
Practice Quiz
Choose an option to reveal the explanation.
1. What does the negative sign in Faraday's law of electromagnetic induction ($e = -N \frac{d\Phi}{dt}$) signify?
Explanation: The negative sign represents Lenz's Law, indicating that the polarity of the induced EMF opposes the change in magnetic flux that produces it.
2. The self-inductance $L$ of a long solenoid of length $l$, cross-sectional area $A$, and total turns $N$ is proportional to which of the following?
Explanation: From the formula $L = \frac{\mu_0 \mu_r N^2 A}{l}$, self-inductance is directly proportional to the square of the number of turns ($N^2$).
3. A metal rod of length $l$ moves with a velocity $v$ perpendicular to a uniform magnetic field $B$. What is the magnitude of the motional EMF induced across its ends?
Explanation: When a conductor moves perpendicular to a uniform magnetic field, the motional EMF generated across its ends is given by $e = Blv$.
4. Lenz's law of electromagnetic induction is a consequence of which fundamental conservation law?
Explanation: Lenz's law ensures that mechanical work done in moving a conductor through a magnetic field transforms into electrical energy, satisfying the law of conservation of energy.
5. The magnetic potential energy stored in an inductor of inductance $L$ carrying a steady current $I$ is given by:
Explanation: The energy stored in the magnetic field of an inductor is $U = \frac{1}{2} LI^2$, analogous to the kinetic energy $\frac{1}{2}mv^2$ in mechanics.
6. What is the SI unit of magnetic flux?
Explanation: Magnetic flux is measured in Weber (Wb), where $1 \text{ Wb} = 1 \text{ T}\cdot\text{m}^2$.
7. If the current in a coil changes at a rate of $2 \text{ A/s}$ and induces a back EMF of $10 \text{ V}$, what is the self-inductance of the coil?
Explanation: Using $|e| = L \frac{dI}{dt}$, we get $10 = L \times 2$, which gives $L = 5 \text{ H}$.
8. The magnetic energy density ($u$) in a uniform magnetic field $B$ in free space is expressed as:
Explanation: Magnetic energy density, or energy stored per unit volume in a magnetic field, is given by $u = \frac{B^2}{2 \mu_0}$.
9. Two coils have self-inductances $L_1$ and $L_2$, and the maximum mutual inductance between them is $M$. The coefficient of coupling $k$ is given by:
Explanation: The coupling coefficient is defined as $k = \frac{M}{\sqrt{L_1 L_2}}$, with values ranging between $0$ and $1$.
10. Which of the following parameters does NOT affect the self-inductance of a given air-core coil?
Explanation: Self-inductance depends only on the physical geometry, core material permeability, and turns of the coil, not on the current flowing through it.
11. In PGT advanced analysis, differential form of Faraday's law links induced electric field $\vec{E}$ with magnetic field $\vec{B}$ as:
Explanation: Maxwell-Faraday equation in differential form is $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$.
12. Eddy currents circulating in bulk metallic conductors subjected to changing magnetic fields cause:
Explanation: Eddy currents flow in closed loops within conductors, generating Joule heating losses which engineers minimize using laminated cores.
13. The principle of betatron relies heavily on which of the following electromagnetic phenomena?
Explanation: A betatron is a particle accelerator that accelerates electrons by means of electromagnetic induction from a time-varying magnetic flux.
14. If the magnetic flux linked with a coil is given by $\Phi = 3t^2 + 5t + 9$, what is the magnitude of the induced EMF at $t = 2 \text{ s}$?
15. What happens to the mutual inductance between two coaxial coils if the distance between them is increased?
Explanation: As the separation between coils increases, magnetic flux linkage per unit primary current decreases, leading to a smaller mutual inductance.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A circular coil of radius $r$ with $N$ turns is rotated with angular velocity $\omega$ in a uniform magnetic field $B$. The peak induced EMF is:
Explanation: Magnetic flux is $\Phi = N B A \cos(\omega t) = N B (\pi r^2) \cos(\omega t)$. Differentiating with respect to time gives peak EMF $e_0 = N B \pi r^2 \omega$.
2. The self-inductance of a coil is $2 \text{ H}$. If a current $I = 4 - 2t$ flows through it, the induced EMF across the coil as a function of time is:
3. A 100-turn coil has a self-inductance of $10 \text{ mH}$. What is the magnetic flux linked with each turn of the coil when a current of $2 \text{ A}$ flows through it?
Explanation: Total flux $\Phi_{\text{total}} = L I = 10 \times 10^{-3} \text{ H} \times 2 \text{ A} = 0.02 \text{ Wb}$. Flux per turn is $\Phi = \frac{\Phi_{\text{total}}}{N} = \frac{0.02}{100} = 2 \times 10^{-4} \text{ Wb}$.
4. Two neighboring coils have a mutual inductance of $1.5 \text{ H}$. If the current in the primary drops from $5 \text{ A}$ to zero in $0.05 \text{ s}$, the average EMF induced in the secondary coil is:
5. A conducting square loop of side $l$ and resistance $R$ is moved out of a uniform magnetic field $B$ at a constant velocity $v$. The force required to maintain this constant velocity is:
Explanation: Induced EMF is $Blv$, current is $I = \frac{Blv}{R}$. Magnetic force opposing motion is $F = I l B = \frac{B^2 l^2 v}{R}$. An equal and opposite external force is needed.
6. What is the total magnetic energy stored in a solenoid of self-inductance $5 \text{ H}$ when a steady current of $4 \text{ A}$ passes through it?
Attempt these questions under a ten-minute limit using a separate timer.
1. An airplane with a wing span of $20 \text{ m}$ is flying horizontally at a speed of $360 \text{ km/h}$ over a region where the vertical component of Earth's magnetic field is $4 \times 10^{-5} \text{ T}$. What is the induced EMF between the tips of the wings?
2. The magnetic flux linked with a circuit of resistance $10 \ \omega$ varies with time according to $\Phi = 6t^2 - 5t + 1$. The total charge that passes through the circuit in the time interval $t = 0$ to $t = 0.5 \text{ s}$ is:
Explanation: Current $I = \frac{1}{R} \frac{d\Phi}{dt} = \frac{1}{10}(12t - 5)$. Charge $Q = \int_0^{0.5} I dt = \frac{1}{10} [6t^2 - 5t]_0^{0.5} = \frac{1}{10}(1.5 - 2.5) = -0.1 \text{ C}$, magnitude is $0.05 \text{ C}$ evaluated correctly between limits.
3. A coil of wire having inductance $L$ and resistance $R$ is connected to a battery of EMF $E$. The time constant of the growth of current in the circuit is:
Explanation: In an $LR$ growth circuit, the time constant $\tau$ is given by the ratio $\frac{L}{R}$.
4. If two identical coils of self-inductance $L$ are connected in series opposition (with mutual inductance $M$), the equivalent inductance is:
Explanation: For series opposition, equivalent inductance is $L_{\text{eq}} = L_1 + L_2 - 2M$. Since $L_1 = L_2 = L$, $L_{\text{eq}} = 2L - 2M = 2(L - M)$.
5. A circular loop of radius $0.1 \text{ m}$ is placed inside a uniform magnetic field of $0.5 \text{ T}$ normal to the loop. If the field drops to zero in $0.01 \text{ s}$, what is the magnitude of the induced EMF?
6. Which of the following statements about eddy currents is INCORRECT?
Explanation: Eddy currents specifically form closed loops within bulk metallic conductors, not open circuit paths.
7. The core of a transformer is laminated to:
Explanation: Lamination introduces high resistance paths that break up large eddy current loops, significantly reducing thermal energy losses.
8. If current in an inductor increases from zero to $6 \text{ A}$ in $0.3 \text{ s}$ and induces an EMF of $20 \text{ V}$, the inductance of the coil is:
Explanation: $|e| = L \frac{\Delta I}{\Delta t} \implies 20 = L \left(\frac{6}{0.3}\right) = L(20) \implies L = 1.0 \text{ H}$.
9. A magnetic field in a region is given by $B = B_0 \sin(\omega t)$. The induced electric field in a concentric circular path of radius $r$ has a magnitude proportional to:
Explanation: Using Faraday's law in integral form, $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi}{dt} = -A \frac{dB}{dt} = -(\pi r^2) B_0 \omega \cos(\omega t)$. Thus $E$ is proportional to $B_0 \omega \cos(\omega t)$.
10. Two coils have mutual inductance $M = 0.4 \text{ H}$. If the primary current changes by $4 \text{ A}$ in $0.02 \text{ s}$, the peak voltage induced in the secondary is: