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Electricity and Magnetism
Faraday's Law of Electromagnetic Induction
Faraday's law of electromagnetic induction states that a change in magnetic flux linked with a closed circuit induces an electromotive force (emf) in it. The magnitude of the induced emf is directly proportional to the time rate of change of magnetic flux.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Calculation of induced emf, magnetic flux, Lenz's law, and Faraday's law for single-turn and multi-turn coils in uniform magnetic fields.
PGT extension
Integral and differential forms of Faraday's law (Maxwell-Faraday equation), motional emf in non-uniform fields, and rotating coils producing alternating emf.
Prerequisites
Definition and calculation of magnetic flux (Phi = B dot A)
Vector calculus fundamentals (Line integrals and surface integrals)
Lorentz magnetic force on moving charges
Magnetic Flux and Flux Linkage
Magnetic flux through an elemental surface dS is given by the scalar product of magnetic field B and area vector dS.
For a coil of N turns, the total magnetic flux linkage is defined as N times the magnetic flux linked with a single turn.
The SI unit of magnetic flux is Weber (Wb) or Tesla-meter squared (T·m2).
Faraday's First and Second Laws
First Law: Whenever the magnetic flux linked with a closed circuit changes, an electric current (and hence an emf) is induced in it.
Second Law: The magnitude of the induced electromotive force is directly proportional to the rate of change of magnetic flux linkage with time.
The dimensional formula of magnetic flux is [M1 L2 T-2 A-1].
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 mathematically incorporates Lenz's law.
Lenz's law is a direct consequence of the law of conservation of energy applied to electromagnetic induction.
Motional Electromotive Force (Motional emf)
When a conductor of length l moves with a velocity v perpendicular to a uniform magnetic field B, an emf is induced across its ends.
The physical origin of motional emf is the Lorentz magnetic force acting on the free electrons inside the moving conductor.
The induced emf causes a redistribution of charges creating an internal electric field until electrostatic and magnetic forces balance.
Derivations
Derivation of Induced emf from Magnetic Flux Change
Conditions: A closed loop enclosing a time-varying magnetic flux Phi(t).
Let the magnetic flux linked with a single turn of a coil at time t be Phi1 and at time t + dt be Phi2.
The change in flux during time interval dt is given by dPh$i = \phi_{2} - \phi_{1}$.
According to Faraday's second law, the magnitude of induced emf e is proportional to dPhi/dt.
Introducing the negative sign for Lenz's law, the equation becomes $e = -N(dPhi/dt)$ for a coil of N turns.
Result: $e = -N \frac{d\Phi}{dt}$
Derivation of Motional emf for a Rod Moving in a Magnetic Field
Conditions: A straight conductor of length l moving with velocity v perpendicular to a uniform magnetic field B.
Consider a conducting rod of length l moving to the right with velocity v in a uniform magnetic field B pointing into the page.
A free charge q inside the rod experiences a magnetic force $F_m = qvB directed along the length of the rod$.
The work done in moving a charge q from one end of the rod to the other is $W = F_m \cdot l = qvBl$.
Since emf is defined as work done per unit charge, $e = W/q = Blv$.
Result: e = Blv
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Magnetic Flux
$$\Phi = \int B \cdot dS = B A \cos\theta$$
B = magnetic field, A = area, theta = angle between B and area vector Weber (Wb)
Uniform or non-uniform magnetic field over surface A
Faraday's Law (Magnitude and Direction)
$$e = -N \frac{d\Phi}{dt}$$
e = induced emf, N = number of turns, dPhi/dt = rate of change of flux Volt (V)
Any closed circuit enclosing changing magnetic flux
Motional emf
$$e = B l v$$
B = magnetic field, l = length of conductor, v = velocity Volt (V)
B, l, and v are mutually perpendicular
Emf Induced in a Rotating Rod
$$e = \frac{1}{2} B \omega l^2$$
B = magnetic field, omega = angular velocity, l = length of rod Volt (V)
Rod rotating about one end in a perpendicular magnetic field
Emf Induced in a Rotating Coil
$$e = N B A \omega \sin(\omega t)$$
N = turns, B = magnetic field, A = area, omega = angular frequency Volt (V)
Coil rotating with constant angular velocity omega in uniform B
Maxwell-Faraday Equation (Differential Form)
$$\nabla \times E = -\frac{\partial B}{\partial t}$$
E = electric field vector, B = magnetic field vector V/m2
General vector form holding in all reference frames
I = induced current, R = resistance of loop, e = induced emf Ampere (A)
Circuit of constant resistance R
Total Charge Flow through a Loop
$$Q = \int I dt = -\frac{N}{R} \Delta \Phi$$
Q = charge flown, N = turns, R = resistance, Delta Phi = total flux change Coulomb (C)
Resistance R is independent of time
Graphs and experiments
Magnet and Coil Relative Motion Experiment
Draw: A bar magnet moving towards a closed solenoid connected to a galvanometer, showing deflection.
Interpret: A faster approach velocity of the magnet produces a larger galvanometer deflection, confirming that induced emf depends on the rate of change of flux.
Exam use: Tested conceptually to verify polarity and magnitude dependence on speed and direction of motion.
Induced emf vs Time for Rotating Coil
Draw: A sinusoidal curve plotting induced emf e(t) versus time t with peaks at +NωBA and -NωBA.
Interpret: Demonstrates the generation of alternating voltage via electromagnetic induction as the angle between area vector and magnetic field changes continuously.
Exam use: Frequently asked in relation to AC generator principles and phase differences between flux and emf.
Magnetic Flux vs Time Graph
Draw: A non-linear curve of flux Phi versus time t, where the slope at any point represents the instantaneous induced emf.
Interpret: The instantaneous induced emf at any instant is equal in magnitude to the slope of the tangent drawn to the Phi-t curve at that instant.
Exam use: Numerical graphical analysis where students must calculate induced emf from slope values.
Solved numericals
TGT core
A circular coil of 500 turns and mean diameter 0.1 m is placed in a uniform magnetic field of 0.4 T normal to the coil plane. If the magnetic field is reduced to zero in 0.02 seconds, find the average induced emf in the coil.
A metal rod of length 1.0 m rotates with a uniform angular velocity of 20 rad/s about an axis passing through one end and perpendicular to the rod. The rod is immersed in a uniform magnetic field of 0.5 T directed parallel to the axis of rotation. What is the induced emf across the ends of the rod?
Identify the orientation: The magnetic field B is parallel to the axis of rotation.
The component of the magnetic field perpendicular to the plane of rotation (or perpendicular to the length and velocity vector) determines the motional emf.
Here, the magnetic field vector is parallel to the rotation axis, meaning the angle between B and the area swept by the rod is 0 degrees, but the velocity vector of any point on the rod is perpendicular to B and the radial vector.
Wait, let's re-verify: If B is parallel to the axis of rotation, B is perpendicular to the plane swept by the rod. Therefore, the magnetic field is perpendicular to the length and velocity of every element of the rod.
Formula for rotating rod: $e = 0.5 * B * \omega * l^2 = 0.5 * 0.5 * 20 * (1.0)^2 = 5.0 V$.
Answer: 5.0 V
PGT extension
The magnetic flux passing through a circuit of resistance 10 ohms varies with time t according to the relation Ph$i = 6t^2 - 5t + 1$, where Phi is in Webers and t is in seconds. Find the total charge that passes through any cross-section of the circuit during the time interval t = 0 to t = 2 seconds.
Find the expression for induced current: I(t) = (1/R) * e(t) = -(1/R) * (dPhi/dt).
Differentiate flux with respect to time: dPhi/d$t = d/dt (6t^2 - 5t + 1) = 12t - 5$.
Current I(t) = - (1/10) * (12t - 5).
Integrate current from t = 0 to t = 2 to find total charge Q = integral of I(t) dt from 0 to 2.
Faraday's law in its elementary algebraic form assumes lumped parameters and uniform fields over macroscopic areas.
The quasi-stationary approximation is assumed, meaning displacement currents are neglected in standard electromagnetic induction derivations.
Common misconceptions
Common error: A stationary conductor in a uniform, time-constant magnetic field can have an induced emf if placed at an angle.
Correct: No emf is induced in a stationary conductor unless the magnetic flux changes with time.
Induction strictly requires a time rate of change of magnetic flux linked with the circuit.
Common error: Lenz's law opposes the magnetic flux itself.
Correct: Lenz's law opposes the *change* in magnetic flux, not the magnetic flux itself.
If flux is constant (even if large), no induced emf is produced.
Common error: Induced emf depends on the resistance of the loop.
Correct: Induced emf depends only on the rate of change of magnetic flux; resistance affects only the induced current.
Emf is the source term (open-circuit voltage); current is a derived quantity determined by Ohm's law ($I = e/R)$.
Quick revision
Induced emf magnitude is proportional to the rate of change of flux linkage: $e = -N(dPhi/dt)$.
Magnetic Flux: \Phi = \int B \cdot dS = B A \cos\theta
Faraday's Law (Magnitude and Direction): $e = -N \frac{d\Phi}{dt}$
Must remember
Magnetic flux through an elemental surface $dS$ is given by $\Phi = \int B \cdot dS = B A \cos\theta$.
Faraday's second law states that magnitude of induced emf is $e = -N \frac{d\Phi}{dt}$.
The negative sign in Faraday's law represents Lenz's law, which is a consequence of conservation of energy.
Motional emf for a straight conductor of length $l$ moving with velocity $v$ perpendicular to field $B$ is $e = Blv$.
Emf induced in a rod rotating about one end in a perpendicular magnetic field is $e = \frac{1}{2}B\omega l^2$.
Maxwell-Faraday differential equation is given by $\nabla \times E = -\frac{\partial B}{\partial t}$.
Total charge flown through a loop of resistance $R$ for a total flux change $\Delta\Phi$ is $Q = \frac{N\Delta\Phi}{R}$.
The SI unit of magnetic flux is Weber (Wb) or $\text{T}\cdot\text{m}^2$.
Glossary
Magnetic Flux ($\Phi$)The total number of magnetic field lines passing normally through a given area.
Induced emf ($e$)The electromotive force produced across a conductor or circuit due to a changing magnetic flux.
Motional emfThe emf induced in a conductor due to its motion through a magnetic field.
Lenz's LawLaw stating that induced current opposes the change in magnetic flux that produces it.
Faraday's First Law vs Second Law
Feature
First Law
Second Law
Nature
Qualitative description of electromagnetic induction
Quantitative measurement of induced emf
Condition
Change in magnetic flux linked with circuit
Rate of change of magnetic flux linkage
Mathematical Expression
Implies $e \neq 0$ when $\frac{d\Phi}{dt} \neq 0$
$e = -N \frac{d\Phi}{dt}$
Practice Quiz
Choose an option to reveal the explanation.
1. What is the SI unit of magnetic flux?
Explanation: The SI unit of magnetic flux is the Weber (Wb), which is equivalent to $\text{T}\cdot\text{m}^2$.
2. According to Faraday's second law, the magnitude of induced emf is proportional to:
Explanation: Faraday's second law states that the magnitude of induced emf is directly proportional to the time rate of change of magnetic flux linkage.
3. The negative sign in Faraday's law equation $e = -N \frac{d\Phi}{dt}$ is a mathematical expression of:
Explanation: The negative sign incorporates Lenz's law, indicating that the induced emf opposes the change in magnetic flux.
4. Lenz's law of electromagnetic induction corresponds to the law of conservation of:
Explanation: Lenz's law is a direct consequence of the law of conservation of energy; otherwise, perpetual motion machines could be created.
5. A metal rod of length $l$ moves with velocity $v$ perpendicular to a uniform magnetic field $B$. The motional emf induced across its ends is:
Explanation: The motional emf across a conductor of length $l$ moving at velocity $v$ perpendicular to $B$ is given by $e = Blv$.
6. What is the dimensional formula of magnetic flux?
Explanation: Magnetic flux $\Phi = B \cdot A$. Since $B = F/(IL)$, dimensions of $B$ are $[M^1 T^{-2} A^{-1}]$, multiplied by area $[L^2]$ gives $[M^1 L^2 T^{-2} A^{-1}]$.ointers.
7. A rod of length $l$ rotates with angular velocity $\omega$ about one of its ends in a perpendicular magnetic field $B$. The induced emf is:
Explanation: Integrating the elemental motional emf $de = B(dr)v = B\omega r dr$ from $0$ to $l$ yields $e = \frac{1}{2}B\omega l^2$.
8. The differential form of Faraday's law (Maxwell-Faraday equation) is expressed as:
Explanation: The Maxwell-Faraday equation in differential form is $\nabla \times E = -\frac{\partial B}{\partial t}$.
9. If a coil has $N$ turns and resistance $R$, and the magnetic flux changes by $\Delta\Phi$, the total charge $Q$ flowing through the coil is given by:
Explanation: Total charge $Q = \int I dt = \int \frac{e}{R} dt = -\frac{N}{R} \int d\Phi = -\frac{N\Delta\Phi}{R}$.
10. Which of the following statements is a common misconception about induced emf?
Explanation: A stationary conductor in a time-constant field has zero induced emf regardless of its orientation because $\frac{d\Phi}{dt} = 0$.
11. A circular coil of area $A$ and $N$ turns rotates with constant angular velocity $\omega$ in a uniform magnetic field $B$. The peak induced emf is:
Explanation: The instantaneous emf is $e = NBA\omega \sin(\omega t)$, so the peak (maximum) emf is $e_0 = NBA\omega$.
12. Can an induced electric field be conservative?
Explanation: Unlike electrostatic fields, induced electric fields associated with time-varying magnetic fields are non-conservative, having non-zero line integrals around closed loops.
13. What happens to the induced current in a loop if its resistance is doubled while keeping the rate of flux change constant?
Explanation: Since induced emf $e$ remains constant and $I = e/R$, doubling $R$ cuts the induced current in half.
14. Magnetic flux linkage through a coil is measured in:
Explanation: For an $N$-turn coil, flux linkage is $N\Phi$, expressed in Weber-turns (or simply Webers since turn is dimensionless).
15. Which physical quantity has the same unit as induced emf?
Explanation: Induced emf is measured in Volts, which is the same unit as electric potential difference.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A flexible circular loop of wire with radius $r$ is placed in a uniform magnetic field $B$ perpendicular to the plane of the loop. If the radius shrinks at a constant rate $\frac{dr}{dt}$, what is the magnitude of the induced emf?
Explanation: Flux $\Phi = B(\pi r^2)$. Magnitude of emf $e = \frac{d\Phi}{dt} = B\pi \frac{d}{dt}(r^2) = 2\pi r B \frac{dr}{dt}$.
2. A rectangular loop of sides $a$ and $b$ is pulled out of a uniform magnetic field $B$ with velocity $v$ at right angles to the field. The resistance of the loop is $R$. What is the induced current in the loop?
Explanation: Only one side of length $b$ is cutting the field while moving at velocity $v$, generating motional emf $e = Bbv$. Thus, current $I = \frac{Bbv}{R}$.
3. A square loop of side $l$ and resistance $R$ is moved out of a magnetic field $B$ with a constant velocity $v$. The total charge flowing through the loop during the time it is pulled completely out of the field is:
4. A wheel with 10 metallic spokes, each of length $1\text{ m}$, rotates with an angular speed of $120\text{ rpm}$ in a plane normal to a horizontal magnetic field of $0.5\text{ T}$. The induced emf between the axle and the rim of the wheel is:
Explanation: Spokes are in parallel, so emf equals that of a single rod. $\omega = 120 \times \frac{2\pi}{60} = 4\pi\text{ rad/s}$. $e = \frac{1}{2}B\omega l^2 = \frac{1}{2}(0.5)(4\pi)(1)^2 = \pi \approx 3.14\text{ V}$.
5. The magnetic flux through a circuit of resistance $R$ changes with time $t$ according to the relation $\Phi = at^2 - bt + c$. The total charge that flows through the circuit from $t = 0$ to $t = T$ is:
Explanation: Initial flux at $t=0$ is $\Phi_1 = c$. Final flux at $t=T$ is $\Phi_2 = aT^2 - bT + c$. Change $\Delta\Phi = aT^2 - bT$. Charge $Q = \frac{\Delta\Phi}{R} = \frac{aT^2 - bT}{R}$.
6. An emf of $2\text{ V}$ is induced in a coil when current in it changes from $2\text{ A}$ to $0\text{ A}$ in $0.1\text{ s}$. What is the self-inductance of the coil?
Explanation: Using magnitude relation $e = L \frac{\Delta I}{\Delta t}$, we get $2 = L \frac{2 - 0}{0.1} \implies 2 = 20L \implies L = 0.1\text{ H}$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. A circular loop of radius $0.1\text{ m}$ is placed in a uniform magnetic field of $0.5\text{ T}$ normal to the loop. If the magnetic field is abruptly reversed in direction in $0.01\text{ s}$, the average induced emf is:
2. Which of the following equations correctly expresses Faraday's law for a stationary closed loop in a time-varying magnetic field?
Explanation: Integral form of Faraday's law is $\oint E \cdot dl = -\frac{d}{dt} \int B \cdot dS$.
3. A solenoid of length $l$, cross-sectional area $A$, and $N$ turns has a current $I$ flowing through it. If the current drops to zero in time $\Delta t$, the induced emf in a secondary co-axial small coil of $n$ turns placed at its center is proportional to:
Explanation: Magnetic field inside solenoid $B = \mu_0 \frac{N}{l} I$. Flux linked with secondary coil $\Phi_2 = n B A$. Induced emf $e = n A \frac{\Delta B}{\Delta t} \propto \frac{N n A I}{l \Delta t}$.
4. A metallic ring is dropped vertically down through a magnetic field region pointing horizontally. As it enters the region, the direction of induced current viewed from above is:
Explanation: As it enters, downward flux increases. By Lenz's law, induced current creates an upward magnetic field opposing entry. Looking from above, an upward magnetic field corresponds to a clockwise current (right-hand grip rule).
5. An aircraft with a wingspan of $20\text{ m}$ flies horizontally at a speed of $360\text{ km/h}$ over a location where the vertical component of Earth's magnetic field is $5 \times 10^{-5}\text{ T}$. The induced emf between the wing tips is:
7. A uniform magnetic field $B$ exists in a cylindrical region of radius $R$. If $B$ increases at a constant rate $\frac{dB}{dt}$, the magnitude of the induced electric field at a distance $r$ ($r < R$) from the axis is:
8. A horizontal metal rod of length $l$ falls freely under gravity from rest. The potential difference developed between its ends, when its horizontal orientation is perpendicular to a horizontal magnetic field $B$, is:
Explanation: Velocity after falling height $h$ is $v = \sqrt{2gh}$. Motional emf $e = Blv = Bl\sqrt{2gh}$.
9. A coil of 50 turns is pulled in $0.02\text{ s}$ from a magnetic field where total flux is $3.1 \times 10^{-4}\text{ Wb}$ to a place where flux is zero. The average induced emf is: