Electromagnetic Induction — NEET notes

Magnetic Flux

Magnetic flux measures 'how much' magnetic field threads through a surface. It is the gateway concept for all of electromagnetic induction — Faraday's law, Lenz's law, and generators all depend on flux changing.

_B = B A = BA

Magnetic Flux Definition

Magnetic flux is the dot product of the magnetic field vector and the area vector. It quantifies the total field threading a given surface.

Magnetic flux _B through a plane surface of area A in a uniform magnetic field B is defined as the dot product of B and the area vector A (which points along the outward normal to the surface).

_B = B A = BA (NCERT Eq. 6.1)

Here is the angle between B and the normal to the surface — NOT the angle between B and the surface itself. This distinction trips up many students.

• SI unit of magnetic flux → Weber (Wb) = T ^2 = V
• CGS unit → Maxwell; 1 Wb = 10^8 Maxwell
• Flux is a scalar (dot product of two vectors)
• = 0° → = BA (maximum flux, field perpendicular to surface plane)
• = 90° → = 0 (field lies parallel to the surface, no field lines 'thread' through)

Flux Through a Coil

For a coil with N turns, the total flux linkage is N times the flux through a single turn. This is what drives the induced EMF in real devices.

A coil with N turns acts like N copies of the same loop stacked together. Each turn intercepts the same flux _B , so the total flux linkage is N _B .

Total flux linkage = N _B = NBA

• Flux linkage = N _B — it has the same unit (Weber) as flux
• Doubling the number of turns doubles the induced EMF
• In NCERT problems, watch whether they ask for flux through one turn ( _B ) or total flux linkage ( N _B )

Three Ways to Change Flux

Flux can change by changing B, A, or theta. Each method corresponds to a different real-world device.

Since _B = BA , flux changes whenever any of the three quantities changes with time.

• Change B : A magnet approaching a coil increases B at the coil → flux changes → EMF induced (Faraday's original experiment)
• Change A : A conducting rod sliding on rails changes the enclosed area → flux changes → motional EMF
• Change : A coil rotating in a constant field changes continuously → flux changes sinusoidally → AC generator

This classification is key to solving NEET problems quickly. Identify WHICH variable is changing, and you immediately know the setup and the approach.

Faraday's Law of Electromagnetic Induction

The induced EMF in a circuit equals the negative rate of change of magnetic flux through it. This single equation — discovered experimentally — is the foundation of generators, transformers, and wireless charging.

= - d _B dt

Faraday's Experiments

Faraday performed three landmark experiments showing that a changing magnetic flux — not a steady one — induces an EMF.

Experiment 1 (Magnet-Coil): Push a bar magnet into a coil connected to a galvanometer. The galvanometer deflects — showing an induced current. Pull the magnet out and the deflection reverses. Hold the magnet still inside the coil — zero deflection.

Experiment 2 (Coil-Coil): Place two coils close together. Pass a changing current through coil 1 (by connecting/disconnecting a battery). Coil 2 shows an induced current — even though it has no battery. A steady current in coil 1 produces no effect in coil 2.

Experiment 3 (Changing area): Slide a conducting rod along parallel rails in a magnetic field. As the enclosed area changes, the galvanometer shows a current.

• A steady magnetic field (or steady flux) produces NO induced EMF
• Only a changing flux induces an EMF — the faster the change, the larger the EMF
• The induced EMF exists as long as the flux is changing; it vanishes the moment flux becomes constant
• Relative motion matters: moving the magnet toward the coil OR moving the coil toward the magnet both work

Faraday's Law

The magnitude of the induced EMF equals the rate of change of flux linkage. For N turns, each turn contributes.

Faraday's law states: the induced EMF in a coil is equal to the negative rate of change of magnetic flux linkage through the coil.

= - d _B dt (single turn, NCERT Eq. 6.3)

= -N d _B dt (coil of N turns, NCERT Eq. 6.4)

The negative sign is Lenz's law (covered next concept). For calculating the magnitude of EMF, use | | = N | | t when the change is uniform over time t .

• Unit of EMF: Volt (V). Consistent: d dt = Wb s = V s = V
• For uniform change: | | = N | _2 - _1| t
• Induced current (if circuit is closed): I = R where R is total resistance
• Charge flowed: q = N| | R — independent of the time taken

Quick recall

A coil of 200 turns has flux changing from 0.05 Wb to 0.01 Wb in 0.02 s. What is the induced EMF?

| | = N | | t = 200 0.04 0.02 = 400 V.

Common Flux-Change Scenarios

Knowing how to express dPhi/dt for different setups is the key skill. Most NEET problems fall into a few standard patterns.

Since _B = BA , differentiation gives three standard cases depending on what changes with time.

• B changes: = -NA dB dt — e.g., coil in a time-varying field
• A changes: = -NB dA dt — e.g., rod sliding on rails
• changes: = NBA ( t) — e.g., coil rotating in a field (AC generator)
• Multiple change: Rarely, two quantities change simultaneously — use the full product rule

Lenz's Law

Lenz's law gives the direction of the induced EMF: it always opposes the change in flux that caused it. This is nature's way of conserving energy — you can't get free electricity from nothing.

= - d _B dt

Lenz's Law Statement

The direction of induced current is such that it opposes the change in magnetic flux that produced it. The negative sign in Faraday's law IS Lenz's law.

Lenz's law states: The polarity of the induced EMF is such that it tends to produce a current which opposes the change in magnetic flux that produced it. This is expressed by the negative sign in Faraday's law.

= - d _B dt

The negative sign is not just a convention — it encodes the direction. If flux is increasing ( d /dt > 0 ), the EMF is negative (opposing the increase). If flux is decreasing ( d /dt < 0 ), the EMF is positive (opposing the decrease).

• Lenz's law is a consequence of conservation of energy
• If the induced current aided the change (instead of opposing), we'd get perpetual motion — violating energy conservation
• The induced current creates its own magnetic field that opposes the change in flux
• Lenz's law determines the direction of induced current; Faraday's law gives the magnitude of EMF

Lenz's Law Applied

The classic example: push a north pole toward a coil and the coil acts like a north pole to repel it. Pull it away and the coil acts like a south pole to attract it.

Case 1: North pole approaches the coil. Flux through the coil increases. The induced current must create a magnetic field that opposes this increase — so the face of the coil facing the magnet becomes a North pole (repels the approaching magnet).

Case 2: North pole moves away from the coil. Flux decreases. The induced current must oppose the decrease — so the face of the coil facing the magnet becomes a South pole (attracts the receding magnet, trying to maintain flux).

• Approaching magnet → coil repels it (induced pole = same as approaching pole)
• Receding magnet → coil attracts it (induced pole = opposite of receding pole)
• Use right-hand curl rule: curl fingers in the direction of current → thumb gives the direction of the magnetic field created by the induced current
• The mechanical energy you spend pushing/pulling the magnet is what gets converted to electrical energy — this is why Lenz's law is conservation of energy

Step-by-Step Strategy

A systematic 4-step method that works for any Lenz's law problem in NEET.

1. Determine the direction of the external magnetic field through the loop/coil
2. Determine whether flux is increasing or decreasing (is the field getting stronger? Is the area growing?)
3. The induced magnetic field must oppose the change: if flux increases, induced B opposes the external B; if flux decreases, induced B supports the external B
4. Use right-hand curl rule on the induced B direction to find the direction of the induced current

Quick recall

A circular loop lies in a region where a uniform magnetic field points into the page and is increasing in magnitude. What is the direction of the induced current?

Flux into the page is increasing. Induced field must oppose the increase, so it points out of the page. By right-hand curl rule, the induced current flows anticlockwise (when viewed from the front).

Motional EMF

When a conductor moves through a magnetic field, the free charges inside it experience a force that creates a potential difference across its ends. This 'motional EMF' is the microscopic engine behind generators and sliding-rail problems.

= Bvl

Straight Conductor Moving in a Field

A rod of length l moving with velocity v perpendicular to a magnetic field B develops an EMF = Bvl across its ends.

Consider a straight conducting rod of length l moving with velocity v perpendicular to a uniform magnetic field B . Every free electron in the rod experiences a Lorentz force F = qvB , pushing electrons toward one end.

Electrons accumulate at one end, creating a potential difference. This continues until the electric field force ( qE ) inside the rod balances the magnetic force ( qvB ). At equilibrium: E = vB , and the potential difference across length l is:

= Bvl (NCERT Eq. 6.5)

• The three vectors v , B , and l must be mutually perpendicular for = Bvl
• If v makes angle with B : = Bvl (only the perpendicular component matters)
• The rod acts as an EMF source (like a battery) in the circuit
• This result can also be derived from Faraday's law: = - d dt = -B dA dt = -Blv (magnitude Bvl )

Sliding Rod on Rails

When the rod slides on conducting rails in a magnetic field, the motional EMF drives a current through the circuit. This leads to power dissipation and a retarding force.

A rod of length l slides on two parallel rails separated by distance l , in a field B perpendicular to the rails. The rails are connected by a resistance R . The rod moves with velocity v .

= Bvl, I = Bvl R , P = B^2v^2l^2 R

The current-carrying rod in a magnetic field also experiences a retarding force (by Lenz's law, opposing the motion):

F_ retarding = BIl = B^2l^2v R

• External force needed to maintain constant velocity = retarding force = B^2l^2v R
• Power delivered by external force = power dissipated in R : Fv = B^2l^2v^2 R — energy is conserved
• If no external force, the rod decelerates — its kinetic energy converts to heat in R
• Charge flowed when rod moves distance d : q = Bld R (independent of speed profile)

Rotating Rod

A conducting rod rotating in a magnetic field sweeps out area at a rate that produces a specific EMF. This appears in wheel/spoke problems.

A conducting rod of length l rotates about one end with angular velocity in a uniform magnetic field B perpendicular to the plane of rotation. Different parts of the rod move at different speeds — the tip moves fastest.

The EMF is found by integrating the motional EMF contribution from each element dx at distance x from the pivot, moving at speed v = x :

= _0^l Bv dx = _0^l B(x ) dx = 1 2 B l^2

• This is the same as the area swept per unit time: dA dt = 1 2 l^2 , so = B 1 2 l^2
• For a metallic disc of radius R spinning at : EMF between centre and rim = 1 2 B R^2
• Multiple spokes in a wheel are all in parallel — net EMF = EMF of one spoke

Quick recall

A wheel with 12 metallic spokes, each 0.5 m long, rotates at 300 rpm in a 0.4 T field. What is the EMF between the axle and the rim?

= 300 2 60 = 10 rad/s. EMF per spoke = 1 2 0.4 10 0.25 = 0.5 1.57 V. All spokes in parallel → net EMF = 0.5 V.

Eddy Currents

When a bulk conductor experiences a changing magnetic flux, currents swirl through the conductor's volume in closed loops — these are eddy currents. They cause heating and braking, and are both a nuisance and a useful tool.

Eddy current dB dt conductor area

What Are Eddy Currents?

When a conducting plate or block moves through a magnetic field (or sits in a changing field), induced currents flow in closed loops within the bulk of the conductor.

Eddy currents (also called Foucault currents) are loops of induced current that flow within the body of a conductor when it is exposed to a changing magnetic field. They are a direct consequence of Faraday's law applied to the bulk of the conductor.

The magnitude of eddy currents depends on: (1) the rate of change of flux, (2) the resistivity of the conductor (low resistivity → large eddy currents), and (3) the geometry — larger conducting area means larger current loops.

• Eddy currents obey Lenz's law — they always oppose the change that causes them
• They flow in closed loops within the conductor's plane (perpendicular to the field)
• The heat produced by eddy currents = I^2R — this is why moving metal in a magnetic field gets hot
• First demonstrated by Leon Foucault (1855) — hence also called Foucault currents

Applications of Eddy Currents

Eddy currents are exploited in electromagnetic braking, induction furnaces, speedometers, and even induction cooktops.

• Electromagnetic braking: Eddy currents in a rotating disc oppose its motion → smooth, contactless braking (used in trains, roller coasters, and laboratory balances)
• Induction furnace: Large eddy currents in a metal piece produce I^2R heating → metal melts without direct flame (used in steel manufacturing)
• Electromagnetic damping: The swinging coil of a galvanometer is wound on a metallic frame so eddy currents damp oscillations quickly → needle settles fast
• Induction cooktop: A coil under the cooktop produces a changing magnetic field → eddy currents heat the metal pan directly (the cooktop surface stays cool)
• Speedometer (car): A rotating magnet induces eddy currents in an aluminum drum → the drum experiences a torque proportional to speed → the needle deflects

Minimizing Eddy Currents

In transformers and motors, eddy currents waste energy as heat. Laminating the core into thin, insulated sheets dramatically reduces them.

Eddy currents are wasteful in transformer and motor cores because they dissipate energy as heat (called eddy current losses). To minimize them, the core is built from thin laminated sheets insulated from each other with varnish.

Lamination works by breaking up the large current loops into tiny ones confined to each thin sheet. Since power loss I^2 and the current scales with the area of the loop, reducing the loop area drastically reduces the loss.

• Transformer cores are made of laminated soft iron sheets (not one solid block)
• Each lamination is coated with insulating varnish or oxide layer
• Slotting (cutting slots in the conductor) also disrupts large eddy current loops
• Using high-resistivity materials (e.g., silicon steel) further reduces eddy current magnitude

Lamination is a NEET favourite. If a question asks 'why are transformer cores laminated?', the answer is always: to reduce eddy current losses.

Self-Induction

A coil carrying a changing current induces an EMF in itself that opposes the change. Self-inductance L measures how strongly a coil resists changes in its own current — it is the electrical analogue of inertia.

= -L dI dt

Self-Inductance L

Self-inductance is the ratio of flux linkage to the current producing it. A higher L means the coil more strongly opposes changes in its current.

When current I flows through a coil, it creates a magnetic field that produces flux linkage N _B through the coil itself. The self-inductance L is defined as the ratio of this flux linkage to the current.

L = N _B I (NCERT Eq. 6.12)

If the current changes, the flux linkage changes, and by Faraday's law, an EMF is induced in the coil itself — called the back-EMF or self-induced EMF.

= -L dI dt (NCERT Eq. 6.13)

• SI unit of inductance → Henry (H) = Wb A = V A
• Named after Joseph Henry who independently discovered electromagnetic induction
• The negative sign means the self-induced EMF opposes the change in current (Lenz's law)
• L depends on geometry (turns, area, length) and core material — NOT on the current itself

Self-Inductance of a Solenoid

A long solenoid of N turns, length l, cross-sectional area A has a simple formula for L. This is the most common geometry in problems.

For a long solenoid with N total turns, length l , cross-sectional area A , and turns per unit length n = N/l :

The magnetic field inside the solenoid is B = _0 nI . Flux through each turn = BA = _0 nIA . Total flux linkage = N _0 nIA = _0 n^2 Al I .

L = N _B I = _0 n^2 Al = _0 N^2 A l (NCERT Eq. 6.15)

• L N^2 — doubling turns quadruples the inductance
• L A — larger cross-section means more inductance
• L 1/l — shorter solenoid (more tightly wound) has higher inductance
• With a core of relative permeability _r : L = _0 _r n^2 Al — iron core boosts L enormously

Quick recall

A solenoid of 1000 turns, 50 cm length, and 10 cm ^2 cross-section. Find L.

L = _0 N^2 A l = 4 10^ -7 10^6 10^ -3 0.5 = 4 10^ -4 0.5 2.5 10^ -3 H = 2.5 mH.

Energy Stored in an Inductor

An inductor carrying current I stores energy in its magnetic field. This energy is released when the current decreases.

When current builds up in an inductor, the self-induced EMF opposes the increase. The external source must do work against this back-EMF. This work is stored as energy in the magnetic field of the inductor.

U = 1 2 LI^2 (NCERT Eq. 6.16)

• Energy is stored in the magnetic field, not in the wire
• Energy density (energy per unit volume) in a magnetic field: u = B^2 2 _0
• When current is switched off, this energy must go somewhere — it appears as the spark at the switch
• For a solenoid: U = 1 2 _0 n^2 I^2 (Al) = B^2 2 _0 volume

Mutual Induction

When changing current in one coil induces an EMF in a nearby coil, this is mutual induction. The mutual inductance M measures the coupling between two coils — it is the principle behind every transformer on Earth.

_2 = -M dI_1 dt

Mutual Inductance M

Mutual inductance is the ratio of flux linkage in one coil to the current in the other. It measures how efficiently one coil can induce EMF in the other.

Consider two coils placed close to each other. When current I_1 flows in coil 1, it creates a magnetic field. Some of this field's flux links with coil 2. The mutual inductance M is defined as:

M = N_2 _ 21 I_1 (NCERT Eq. 6.9)

Here _ 21 is the flux through each turn of coil 2 due to the current I_1 in coil 1. If I_1 changes, the EMF induced in coil 2 is:

_2 = -M dI_1 dt (NCERT Eq. 6.10)

• SI unit of M → Henry (H) — same as self-inductance
• M depends on: geometry, number of turns of both coils, relative orientation, separation, and core material
• More overlap of flux between the coils → larger M
• If the coils are perpendicular (zero flux coupling), M = 0

Reciprocity Theorem

The mutual inductance M12 = M21 = M. Whether coil 1 induces in coil 2 or coil 2 induces in coil 1, the coupling constant is the same.

A remarkable property of mutual inductance: M_ 12 = M_ 21 = M . The flux linkage in coil 2 per unit current in coil 1 is exactly equal to the flux linkage in coil 1 per unit current in coil 2.

M_ 12 = N_2 _ 21 I_1 = M_ 21 = N_1 _ 12 I_2 = M

• Reciprocity means we only need ONE value of M for any pair of coils
• This is why transformer turns ratio works symmetrically
• For problems: calculate M from whichever direction is easier, then use it for either direction

M for Two Coaxial Solenoids

Two solenoids wound on the same axis (one inside the other) have a simple mutual inductance formula. This is the most tested case.

Consider a solenoid S_1 (turns N_1 , length l , area A_1 ) with a shorter solenoid S_2 (turns N_2 , area A_2 ) wound tightly over it. The inner solenoid's field is B_1 = _0 n_1 I_1 (where n_1 = N_1/l ).

Flux through each turn of S_2 due to S_1 : _ 21 = B_1 A_1 = _0 n_1 I_1 A_1 (using A_1 because the field is confined to the inner solenoid's cross-section).

M = N_2 _ 21 I_1 = _0 n_1 N_2 A_1 = _0 N_1 N_2 A_1 l

• M N_1 N_2 — more turns on either coil increases the coupling
• M uses the area of the inner solenoid (where the field exists)
• With a core of relative permeability _r : M = _0 _r n_1 N_2 A_1
• Relation between M , L_1 , L_2 : M = k L_1 L_2 where k = coupling coefficient ( 0 k 1 )

Quick recall

Two coaxial solenoids: inner has 400 turns, area 25 cm ^2 ; outer has 700 turns; common length = 40 cm. Find M.

M = _0 N_1 N_2 A l = 4 10^ -7 400 700 25 10^ -4 0.4 = 4 10^ -7 2.8 10^5 2.5 10^ -3 0.4 2.2 10^ -4 H = 0.22 mH.

AC Generator

An AC generator converts mechanical energy into alternating electrical energy by rotating a coil in a magnetic field. The EMF varies sinusoidally — this is the device that powers the entire electrical grid.

= NBA ( t)

Working Principle

As a coil rotates in a uniform magnetic field, the angle between B and the area vector changes continuously, producing a sinusoidally varying EMF.

A rectangular coil of N turns, area A , rotates with angular velocity in a uniform magnetic field B . At time t , the angle between B and the area vector A is = t (starting from the position where B A ).

The flux at time t : = NBA ( t) . By Faraday's law:

= - d dt = NBA ( t) (NCERT Eq. 6.19)

The peak EMF (maximum value) is:

_0 = NBA

• EMF is maximum when the coil plane is parallel to B ( = 90° , flux = 0, rate of change is maximum)
• EMF is zero when the coil plane is perpendicular to B ( = 0° or 180° , flux = max or min, rate of change is zero)
• Frequency of output: f = /2 — India uses 50 Hz, so the coil rotates 50 times per second
• _0 N , B , A , — increase any one to get higher peak voltage

Construction

The essential components: coil (armature), magnetic field (permanent magnet or electromagnet), slip rings, and brushes.

• Armature: The rectangular coil of insulated copper wire wound on a soft iron core (increases flux)
• Field magnet: Strong permanent magnets or electromagnets providing uniform B
• Slip rings: Two metallic rings attached to the two ends of the coil; they rotate WITH the coil
• Brushes: Carbon brushes pressed against the slip rings; they remain stationary and connect to the external circuit
• Shaft/axle: Mechanical input — connected to a turbine (steam/water/wind) that rotates the coil

As the coil rotates, the slip rings rotate with it. The brushes maintain sliding contact with the rings, allowing the alternating EMF to be drawn off to the external load without tangling wires.

AC vs DC Generator

The only structural difference between an AC and DC generator is the output mechanism: slip rings (AC) vs split-ring commutator (DC).

Both AC and DC generators use the same principle (rotating coil in a magnetic field). The coil generates a sinusoidal EMF in both cases. The difference lies in how this EMF is extracted.

• A split-ring commutator is just a slip ring cut in half with a gap
• Every time the EMF would reverse, the commutator swaps connections so external current keeps the same direction
• The DC output is not smooth — it pulsates. A filter capacitor or multiple coils smooth it out

Quick recall

An AC generator has 100 turns, area 0.1 m ^2 , B = 0.5 T, and rotates at 50 Hz. Find the peak EMF.

= 2 50 = 100 rad/s. _0 = NBA = 100 0.5 0.1 100 = 500 1571 V.