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Electricity and Magnetism
Electromagnetic Induction
Electromagnetic induction is the phenomenon of generating an electromotive force (emf) in a circuit by changing the magnetic flux linked with it. It forms the foundational principle underlying electrical generators, transformers, and inductors.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Faraday's laws of induction, Lenz's law, motional emf, self-induction, and mutual induction.
PGT extension
Differential and integral forms of Maxwell's equations relating to electromagnetic induction, betatron principle, eddy currents, and time-varying magnetic fields generating non-conservative electric fields.
Prerequisites
Magnetic field and magnetic flux
Lorentz force and magnetic force on moving charges
Gauss's law for magnetism
Faraday's Laws of Electromagnetic Induction
First Law: Whenever the magnetic flux linked with a closed circuit changes, an electromotive force (emf) is induced in the circuit, which lasts only as long as the change in magnetic flux continues.
Second Law: The magnitude of the induced emf is directly proportional to the time rate of change of magnetic flux linked with the circuit.
Magnetic flux is defined as the scalar product of the magnetic field vector and the area vector, given by $\Phi = B \cdot A = BA \cos\theta$.
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.
Mathematically, this opposition is incorporated as a negative sign in Faraday's law: $e = - (d\Phi / dt)$.
Lenz's law is a direct consequence of the law of conservation of energy; mechanical work done in moving a conductor against a magnetic force is converted into electrical energy.
Motional Electromotive Force
When a conductor of length l moves with a velocity v perpendicular to a uniform magnetic field B, a potential difference is induced across its ends.
The free electrons inside the conductor experience a magnetic Lorentz force ($F = q(v \times B))$, causing them to accumulate at one end until an electrostatic field balances the magnetic force.
The resulting motional emf is given by e = Blv when velocity, magnetic field, and length are mutually perpendicular.
Self-Induction and Mutual Induction
Self-induction is the property of a coil by virtue of which it opposes any change in the strength of current flowing through it by inducing an emf within itself.
The self-induced emf depends on the self-inductance L, such that $e = - L (di / dt)$.
Mutual induction is the phenomenon where a change in current in a primary coil induces an emf in a nearby secondary coil, quantified by the mutual inductance M.
Derivations
Motional Electromotive Force in a Conducting Rod
Conditions: A straight conducting rod of length l moving with constant velocity v in a uniform perpendicular magnetic field B.
Consider a charge q inside the rod moving with velocity v in magnetic field B. The magnetic force on the charge is $F_m = qvB$.
This force drives charges toward one end, establishing an electric field E inside the rod such that the electrostatic force $F_e = qE balances the magnetic force at equilibrium$.
Equating magnitudes: qE = qvB, which gives the internal electric field E = vB.
The induced emf e is the line integral of the electric field along the length l of the rod: $e = \int E dl = E l = Blv$.
Result: e = Blv
Energy Stored in an Inductor
Conditions: A current i is established in an inductor of self-inductance L against the back emf.
The instantaneous back emf opposing the growth of current is $e = L (di / dt)$.
The rate at which work must be done by an external source to drive the current against this back emf is $P = - ie = i L (di / dt)$.
The total work W done in increasing the current from 0 to I is obtained by integrating the power over time: $W = \int P dt = \int L i di from 0$ to I.
Performing the integration yields $W = \frac{1}{2} LI^2$.
Result: $U = \frac{1}{2} LI^2$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Magnetic Flux
$$\Phi = \int B \cdot dA$$
B = magnetic field, A = area vector Weber (Wb) or Tesla meter squared (T m^2)
General definition for uniform or non-uniform fields.
Faraday-Lenz Law
$$e = - \frac{d\Phi}{dt}$$
e = induced emf, \Phi = magnetic flux, t = time Volt (V)
Single-turn loop.
Faraday Law for N-turn Coil
$$e = - N \frac{d\Phi}{dt}$$
N = number of turns, \Phi = flux per turn Volt (V)
Coil with N tightly wound turns.
Motional emf
$$e = Blv$$
B = magnetic field, l = length of conductor, v = velocity Volt (V)
B, l, and v are mutually perpendicular.
Self-Inductance definition
$$N\Phi = Li$$
N\Phi = total flux linkage, $L = self-inductance$, i = current Henry (H)
Linear magnetic medium.
Self-induced emf
$$e = - L \frac{di}{dt}$$
$L = self-inductance$, di/dt = rate of change of current Volt (V)
Constant inductance L.
Mutual Inductance
$$N_2\Phi_2 = M i_1$$
$N_2\Phi_2 = flux linkage in secondary$, M = mutual inductance, $i_1 = primary current$ Henry (H)
magnetically coupled coils.
Magnetic Energy Density
$$u = \frac{B^2}{2 \mu_0}$$
u = energy per unit volume, B = magnetic field, \m$u_0 = permeability of free space$ Joule per cubic meter (J m^-3)
Vacuum or non-magnetic medium.
Graphs and experiments
Magnet and Coil Relative Motion (Faraday's Experiment)
Draw: A bar magnet moving toward a stationary closed loop connected to a zero-center galvanometer, showing deflection.
Interpret: Approaching the north pole deflects the galvanometer in one direction, while receding deflects it oppositely, confirming directional dependence of induced emf (Lenz's law).
Exam use: Tested conceptually in TGT to check polarity prediction and direction of induced current.
Growth of Current in an L-R Circuit
Draw: An exponential growth curve of current i versus time t, starting from 0 and approaching a steady-state value $I_0 = V/R with time constant \tau = L/R$.
Interpret: The back emf slows down the immediate rise of current when a DC voltage is switched on, asymptotic to the maximum ohmic current.
Exam use: Frequently asked in PGT for time constant calculations and graphical interpretation of transient states.
Alternating Magnetic Flux and Induced emf Sine Wave
Draw: Sinusoidal magnetic flux \Phi(t) = \Phi_0 \cos(\omega t) plotted alongside its derivative curve (induced emf) which is a sine wave shifted in phase.
Interpret: Demonstrates that rate of change of a cosine flux wave produces a sine wave emf, leading by 90 degrees in phase.
Exam use: Relates electromagnetic induction directly to AC generation principles in electrical machinery.
Solved numericals
TGT core
A circular loop of radius 0.1 m is placed in a uniform magnetic field of 0.5 T perpendicular to the plane of the loop. If the magnetic field is reduced to zero in 0.02 s, find the magnitude of the induced emf in the loop.
Calculate initial magnetic flux: \Ph$i_1 = B \cdot A = 0.5 \times \pi \times (0.1)^2 = 0.5 \times 3.1416 \times 0.01 = 0.0157 \text{ Wb}$.
Calculate value: e = \frac{0.0157}{0.02} = 0.785 \text{ V}.
Answer: 0.785 V
PGT extension
A metallic rod of length 0.5 m rotates with an angular velocity of 20 rad/s about an axis passing through one of its ends, perpendicular to the rod, inside a uniform magnetic field of 0.4 T directed parallel to the axis of rotation. Wait, let's correct the geometry: magnetic field is perpendicular to the plane of rotation. Calculate the induced emf across the ends of the rod.
Consider an element of the rod of length dr at a distance r from the axis of rotation.
The linear speed of this element is v = \omega r.
The elemental induced emf is de = B v dr = B \omega r dr.
Integrate from r = 0 to $r = l: e = \int_0^l B \omega r dr = \frac{1}{2} B \omega l^2$.
Substitute values: B = 0.4 T, $\omega = 20 \text{ rad/s}$, l = 0.5 \text{ m}.
An inductor of self-inductance 5 H carries a steady current of 2 A. If the current is uniformly interrupted and drops to zero in 10 ms, calculate the average induced emf during this time interval.
Calculate final value: e = 5 \times 200 = 1000 \text{ V}.
Answer: 1000 V
Assumptions and limitations
Faraday's law in the form $e = - d\Phi/dt assumes that the path of integration$ is stationary in inertial frames, whereas moving media require the Lorentz force approach.
Common misconceptions
Common error: Induced emf requires a complete closed circuit to be generated.
Correct: An induced emf is produced across the ends of a conductor whenever magnetic flux changes, regardless of whether the circuit is open or closed.
Current only flows if the circuit is closed, but the emf exists purely due to flux variation or conductor motion.
Common error: Lenz's law violates the conservation of energy because it opposes motion.
Correct: Lenz's law is a direct enforcement of conservation of energy.
External mechanical work must be continuously performed to overcome magnetic braking forces, which directly supplies the electrical energy generated.
Quick revision
The negative sign in Faraday's law mathematically represents Lenz's law and energy conservation.
Magnetic Flux: \Phi = \int B \cdot dA
Faraday-Lenz Law: $e = - \frac{d\Phi}{dt}$
Must remember
Faraday's Second Law states that the magnitude of induced emf is directly proportional to the time rate of change of magnetic flux linked with the circuit, given by $e = -N \frac{d\Phi}{dt}$.
Lenz's law is a direct consequence of the law of conservation of energy, ensuring that the induced emf opposes the change in magnetic flux.
Motional emf for a conductor of length $l$ moving with velocity $v$ perpendicular to a uniform magnetic field $B$ is given by $e = Blv$.
Self-induction is the phenomenon where a change in current in a coil induces a back emf within itself, expressed as $e = -L \frac{di}{dt}$.
Mutual induction involves the flux linkage in a secondary coil due to a changing current in a primary coil, quantified by $N_2\Phi_2 = M i_1$.
Glossary
Magnetic Flux ($\Phi$)The scalar product of the magnetic field vector and the area vector, measured in Weber (Wb).
Self-Inductance ($L$)The property of a coil that opposes any change in its own current, measured in Henry (H).
Mutual Inductance ($M$)The measure of coupling between two magnetically linked coils.
Motional emf ($e$)The potential difference induced across the ends of a moving conductor in a magnetic field.
Energy Density ($u$)The magnetic energy stored per unit volume, given by $u = \frac{B^2}{2\mu_0}$.
Self-Induction vs Mutual Induction
Feature
Self-Induction
Mutual Induction
Number of Coils
Single coil
Two or more coils (Primary and Secondary)
Governing Equation
$e = -L \frac{di}{dt}$
$e_2 = -M \frac{di_1}{dt}$
Primary Cause
Change of current within the same coil
Change of current in a neighboring coil
Practice Quiz
Choose an option to reveal the explanation.
1. What does the negative sign in Faraday's law of electromagnetic induction represented by $e = -\frac{d\Phi}{dt}$ signify?
Explanation: The negative sign represents Lenz's law, which states that the induced emf opposes the change in magnetic flux that produces it, conforming to the conservation of energy.
2. What is the SI unit of magnetic flux?
Explanation: Magnetic flux $\Phi = B \cdot A$ is measured in Weber (Wb), which is equivalent to Tesla meter squared ($T\cdot m^2$). Note that Volt-second is dimensionally identical, but Weber is the standard primary SI unit.
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 induced motional emf?
Explanation: When the velocity, magnetic field, and length are mutually perpendicular, the motional emf induced across the ends of the conductor is given by $e = Blv$.
4. Which law or principle is fundamentally responsible for Lenz's law?
Explanation: Lenz's law is a direct consequence of the law of conservation of energy. If it aided the change, perpetual motion would be created, violating energy conservation.
5. The self-inductance of a coil depends upon:
Explanation: Self-inductance $L$ is a geometric and material property of the inductor, depending on the number of turns, area, length, and permeability of the core, independent of current or its rate of change.
6. What is the magnetic energy density stored in a vacuum where the magnetic field is $B$?
Explanation: The magnetic energy density (energy per unit volume) in a vacuum or non-magnetic medium is given by the formula $u = \frac{B^2}{2\mu_0}$.
7. If the number of turns in a coil is doubled, keeping all other parameters constant, how does the self-inductance change?
Explanation: Self-inductance is proportional to the square of the number of turns ($L \propto N^2$). Therefore, doubling $N$ increases $L$ by a factor of 4.
8. An induced emf is produced whenever:
Explanation: According to Faraday's law, an induced emf is generated whenever there is a change in the magnetic flux linked with the circuit.
9. What is the dimensional formula for self-inductance $L$?
10. Which of the following describes the core principle behind the operation of a betatron (PGT extension)?
Explanation: A betatron accelerates electrons via a changing magnetic flux, which induces a non-conservative electric field according to Faraday's law generalized in Maxwell's equations.
11. Eddy currents are produced when:
Explanation: Eddy currents are circulating currents induced in bulk conductors when they experience a changing magnetic flux, leading to magnetic damping and heating.
12. Which Maxwell equation represents the differential form of Faraday's Law of Induction?
Explanation: The differential form of Faraday's law is $\nabla \times E = -\frac{\partial B}{\partial t}$, showing that a time-varying magnetic field creates a non-conservative electric field.
13. Two coils have a mutual inductance $M = 1.5\text{ H}$. If the current in the primary changes at the rate of $20\text{ A/s}$, what is the magnitude of the induced emf in the secondary coil?
Explanation: The induced emf in the secondary coil is $e_2 = M \frac{di_1}{dt} = 1.5 \times 20 = 30\text{ V}$.
14. A circular loop of radius $r$ is placed in a uniform magnetic field $B$ normal to the plane of the loop. If the radius expands at a constant rate $\frac{dr}{dt}$, what is the induced emf?
Explanation: Here $\Phi = B(\pi r^2)$. The induced emf is $e = -\frac{d\Phi}{dt} = -B\pi \frac{d}{dt}(r^2) = -2\pi r B \frac{dr}{dt}$.
15. What is the primary reason why laminated iron cores are used in transformers?
Explanation: Lamination breaks up the paths for eddy currents within the iron core, substantially reducing power loss due to Joule heating.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A magnetic flux linked with a coil varies with time $t$ as $\Phi = 3t^2 + 5t + 9$. What is the magnitude of the induced emf at $t = 2\text{ s}$?
2. A copper disc of radius $r$ rotates with an angular velocity $\omega$ normal to a uniform magnetic field $B$. What is the induced emf between the center and the rim of the disc?
Explanation: Considering an elemental strip, the motional emf integrated from center to rim yields $e = \frac{1}{2}B\omega r^2$.
3. When the current in a coil changes from $2\text{ A}$ to $10\text{ A}$ in $0.1\text{ s}$, an average emf of $40\text{ V}$ is induced. What is the self-inductance of the coil?
Explanation: Using $|e| = L \frac{\Delta i}{\Delta t}$, we have $40 = L \frac{10 - 2}{0.1} = L \frac{8}{0.1}$, which gives $L = \frac{4}{8} = 0.5\text{ H}$.
4. A rectangular loop of sides $a$ and $b$ is pulled out of a magnetic field $B$ with a constant velocity $v$. If the resistance of the loop is $R$, what is the induced current?
Explanation: As the loop moves out, only one side of length $b$ cuts the magnetic field perpendicularly, producing motional emf $e = Bbv$. The current is $I = \frac{e}{R} = \frac{Bbv}{R}$.
5. What is the mutual inductance between two concentric coplanar circular coils of radii $r_1$ and $r_2$ (where $r_1 \ll r_2$)?
Explanation: The magnetic field at the center due to the outer coil of radius $r_2$ is $B = \frac{\mu_0 i_2}{2 r_2}$. The flux linkage with the inner coil is $\Phi_1 = B (\pi r_1^2) = \frac{\mu_0 \pi r_1^2}{2 r_2} i_2$, so $M = \frac{\mu_0 \pi r_1^2}{2 r_2}$.
6. A solenoid of length $l$, cross-sectional area $A$, and total $N$ turns has a self-inductance given by which expression?
Explanation: For a long solenoid, magnetic field $B = \frac{\mu_0 N i}{l}$. Flux per turn $\Phi = B \frac{A}{N}$ or total flux linkage $N\Phi = B A N = \frac{\mu_0 N^2 A i}{l}$. Thus $L = \frac{N\Phi}{i} = \frac{\mu_0 N^2 A}{l}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. The magnetic flux linked with a coil of $N$ turns changes from $\Phi_1$ to $\Phi_2$ in time $t$. The induced emf is given by:
Explanation: According to Faraday's law for $N$ turns, $e = -N \frac{d\Phi}{dt} = -N \frac{\Phi_2 - \Phi_1}{t}$.
2. A metal bar of length $1\text{ m}$ is rotated about one of its ends in a plane perpendicular to a magnetic field of $0.5\text{ T}$ with a frequency of $50\text{ rev/s}$. What is the induced emf?
Explanation: Angular velocity $\omega = 2\pi f = 2\pi(50) = 100\pi\text{ rad/s}$. The induced emf is $e = \frac{1}{2} B \omega l^2 = \frac{1}{2} (0.5)(100\pi)(1)^2 = 25\pi \approx 78.5\text{ V}$.
3. Which of the following statements is correct regarding self-inductance?
Explanation: Self-inductance opposes any change in current, acting as an 'electrical inertia' in electric circuits.
4. If the current in an inductor of inductance $2\text{ H}$ changes at the rate of $4\text{ A/s}$, the magnitude of the self-induced emf is:
Explanation: Using $e = -L \frac{di}{dt}$, the magnitude is $|e| = 2 \times 4 = 8\text{ V}$.
5. In the context of electromagnetic induction, a non-conservative electric field is generated by:
Explanation: Faraday's law implies that time-varying magnetic fields produce electric fields that are non-conservative (having non-zero line integrals around closed loops).
6. Two coils have self-inductances $L_1 = 4\text{ H}$ and $L_2 = 9\text{ H}$. If the magnetic coupling between them is ideal (coefficient of coupling $k = 1$), what is their mutual inductance $M$?
Explanation: Mutual inductance for coupled coils is given by $M = k\sqrt{L_1 L_2}$. With $k = 1$, $M = \sqrt{4 \times 9} = \sqrt{36} = 6\text{ H}$.
7. A train is moving northwards on rails separated by $1.2\text{ m}$. If the vertical component of Earth's magnetic field is $2 \times 10^{-5}\text{ T}$ and the speed of the train is $72\text{ km/h}$, what is the induced emf between the rails?
8. What happens to the magnetic energy stored in an inductor when the current flowing through it is doubled?
Explanation: The magnetic energy stored in an inductor is given by $U = \frac{1}{2}Li^2$. Doubling the current $i$ increases the stored energy by a factor of $2^2 = 4$.
9. Which experiment conclusively demonstrates electromagnetic induction and Lenz's law simultaneously?
Explanation: Moving a magnet relative to a coil demonstrates induced current and the mechanical resistance felt due to Lenz's law.
10. What is the role of eddy currents in magnetic braking systems of trains?
Explanation: Magnetic braking induces eddy currents in metal wheels or rails, which create opposing magnetic forces (Lenz's law) resulting in smooth, friction-free deceleration.