Electromagnetic Induction

🎯 Key Points

• Faraday's Law: induced EMF = −dΦ/dt, where Φ=B·A·cosθ is the magnetic flux through a circuit - EMF is induced only while flux is CHANGING, never when it's constant
• Lenz's Law: the induced current always flows in a direction that OPPOSES the change in flux that created it - a direct consequence of conservation of energy
• Motional EMF: a rod of length L moving with velocity v perpendicular to field B generates EMF = BLv
• Self-induction EMF = −L(dI/dt); Mutual induction EMF₂ = −M(dI₁/dt)
• An AC generator converts mechanical energy to electrical energy by rotating a coil in a magnetic field, producing EMF = NBAω sin(ωt)

Faraday's Law of Electromagnetic Induction

• Whenever the magnetic flux Φ=B·A·cosθ linked with a circuit changes, an EMF is induced in it, given by EMF = −N(dΦ/dt) for a coil of N turns
• Flux can change because B changes, A changes, the angle θ between B and the area's normal changes, or any combination of these
• No flux change means no induced EMF, no matter how strong the field is - a magnet held stationary near a coil induces nothing

Faraday's law: a changing current in the primary coil changes the magnetic flux through the secondary coil, inducing an emf that deflects the galvanometer. Image: Eviatar Bach, CC0, via Wikimedia Commons.

Lenz's Law and Energy Conservation

• The induced current flows in whichever direction creates a magnetic field that opposes the change in flux - if flux is increasing, the induced current creates a field to oppose the increase; if decreasing, it creates a field to reinforce it
• This opposition means external work must always be done against the induced effects to change the flux - Lenz's law is simply conservation of energy applied to electromagnetism
• The negative sign in Faraday's law, EMF=−dΦ/dt, mathematically encodes Lenz's law

Motional EMF

• A straight conductor of length L moving with velocity v perpendicular to a magnetic field B generates a motional EMF = BLv, caused by the magnetic force on free charges within the moving conductor (an application of the Lorentz force, distinct from flux-based reasoning, though both give the same answer)
• If such a rod slides on conducting rails connected to a resistor, the induced current experiences a retarding force (by Lenz's law) that opposes the motion - leading to a terminal velocity if a constant external force drives the rod, analogous to a viscous drag problem

Self-Induction and Mutual Induction

• Self-induction: a changing current in a coil induces an EMF in the SAME coil, EMF = −L(dI/dt), where L is the self-inductance (henry, H) - this is why circuits with inductors resist sudden changes in current
• Mutual induction: a changing current in one coil induces an EMF in a NEARBY coil, EMF₂ = −M(dI₁/dt), where M is the mutual inductance - the basic principle behind the transformer
• Energy stored in an inductor carrying current I: U = (1/2)LI², stored in the magnetic field itself

Eddy Currents and the AC Generator

• Eddy currents: circulating currents induced in the bulk of a conductor (not confined to a wire) by a changing flux; they oppose the change causing them (used for electromagnetic braking) but cause unwanted heating in transformer/motor cores, minimised by using laminated cores instead of solid metal blocks
• An AC generator converts mechanical energy into electrical energy: a coil of N turns and area A rotates at angular velocity ω in a uniform field B, producing an induced EMF = NBAω sin(ωt) - a sinusoidally alternating voltage

Self-Inductance of a Solenoid

• Self-inductance L depends only on a coil's geometry and its core material, defined through the flux linkage NΦ = LI
• For a long solenoid of N turns, length l, cross-sectional area A, with n = N/l turns per unit length: L = μ₀n²Al = μ₀N²A/l (air core); with a core of permeability μ, replace μ₀ by μ
• L is measured in henry (H): a coil has 1 H if a current changing at 1 A/s induces 1 V across it
• A large inductance opposes rapid current change strongly, so inductors smooth out and resist sudden fluctuations in current

Mutual Inductance of Coupled Coils

• Mutual inductance M relates the flux linked in coil 2 to the current in coil 1: N₂Φ₂ = MI₁, and reciprocally M₁₂ = M₂₁ = M
• For two coaxial solenoids (turns per unit length n₁ and n₂, common length l, area A), M = μ₀n₁n₂Al
• The coupling coefficient k (with 0 ≤ k ≤ 1) gives M = k√(L₁L₂); k = 1 means perfect flux linkage (ideal transformer), while k < 1 indicates flux leakage between the coils

Energy Stored in an Inductor and Magnetic Energy Density

• Building up a current I in an inductor requires work against the back-EMF; this work is stored as magnetic field energy U = (1/2)LI²
• Expressed per unit volume, the magnetic energy density is u = B²/(2μ₀) - the magnetic analogue of the electric energy density (1/2)ε₀E²
• This stored energy is what sustains the spark seen when an inductive circuit is suddenly switched off

🚀 JEE Advanced Edge

Motional EMF and rod-on-rails problems: For a conducting rod of length L sliding with velocity v perpendicular to a magnetic field B, motional EMF = BLv. If the rod moves on frictionless rails connected to a resistor, this EMF drives a current that, by Lenz's law, creates a retarding force on the rod - leading to terminal velocity problems analogous to viscous drag.

Eddy currents: Induced circulating currents in a bulk conductor (not confined to a wire) caused by a changing flux; they oppose motion/change (used for electromagnetic braking) and cause unwanted heating in transformer cores (minimised using laminated cores instead of solid blocks).

Worked problem: A coil of 200 turns and area 0.02 m² rotates at 50 rotations per second in a uniform magnetic field of 0.4 T. Find the peak EMF generated. Approach: ω = 2πf = 2π(50) = 100π rad/s. Peak EMF = NBAω = 200 × 0.4 × 0.02 × 100π = 1.6 × 100π = 160π ≈ 502.7 V.

Magnetic Flux and Its Definition

• Magnetic flux through a surface is phi = B A cos theta, where theta is the angle between B and the area vector (normal) of the surface.
• Its SI unit is the weber (Wb), where 1 Wb = 1 T m squared; flux is a scalar quantity.
• Flux is maximum when the field is perpendicular to the plane of the loop (theta = 0) and zero when the field lies in the plane (theta = 90 degrees).
• For a coil of N turns, the total linked flux is N phi, called the flux linkage.
• A change in flux can be produced by changing B, the area A, or the orientation theta of the loop.

Faraday and Henry's Experiments

• Moving a bar magnet towards or away from a coil connected to a galvanometer induces a current; the deflection reverses when the motion reverses.
• No current flows when the magnet is held stationary, showing that only a changing flux induces an EMF.
• Replacing the magnet with a second current-carrying coil produces the same effect when the current in it is switched on, off, or varied.
• The magnitude of the induced current depends on the rate of relative motion or rate of change of current, not on the flux itself.
• These experiments established that relative change between a magnetic field and a circuit is the cause of induction.

Inductors in Circuits and Their Behaviour

• An inductor opposes changes in current by developing a back EMF equal to minus L times (dI divided by dt).
• Because current in an inductor cannot change instantaneously, an inductor behaves like an open circuit at the instant of switching on.
• In steady DC, dI divided by dt is zero, so the inductor behaves like a plain wire (short circuit) with no voltage drop.
• Inductors in series add: L equivalent = L₁ + L₂ (no mutual coupling); in parallel the reciprocals add.
• The energy delivered while establishing current is stored in the magnetic field as U = one half L I squared.

Induced Electric Field and Its Nature

• A time-varying magnetic field creates an induced electric field even in the absence of any conductor.
• This field is non-conservative: its line integral around a closed loop is not zero but equals minus d(phi) divided by dt.
• Unlike electrostatic fields, the induced field lines form closed loops and are not associated with static charges.
• The induced EMF in a circuit equals the work done by this field per unit charge around the loop.
• This concept explains induction in stationary conductors and underlies Maxwell's generalisation of Faraday's law.