Electromagnetic Induction: theory

CONCEPT

Magnetic Flux & Faraday’s Law

In one line
A change in the magnetic flux through a circuit produces an emf. The faster the flux changes, the larger the emf.

FLUX

12.1.1 Magnetic flux

Number of field lines passing normally through an area.
Scalar quantity.
θ is the angle between B and the area vector (the normal).

φ = B · A = BA cos θ

Area vector A is along the normal

θ = 0° (plane ⊥ field): φ = BA, the maximum.
θ = 90° (plane ∥ field): φ = 0.
Net flux through any closed surface is zero.
If B = B_xî + B_yĵ + B_zk̂ and the area lies in the XY plane, only B_z counts.

UNITS

12.1.2 Unit and dimension

SI unit: weber (Wb) = T m² = V s = H A.
CGS unit: maxwell. 1 Wb = 10⁸ maxwell.
Dimension: [ML²T⁻²A⁻¹].

FARADAY’S LAWS

12.1.3 Faraday’s laws

First law: an emf is induced whenever the flux through a circuit changes. It lasts only as long as the change lasts.
Second law: the emf equals the rate of change of flux linkage.

e = −N dφdt

The negative sign shows opposition (Lenz’s law).
φ is the flux per turn; Nφ is the flux linkage.

average emf: e = −N φ₂ − φ₁Δt

WAYS TO CHANGE FLUX

12.1.4 φ = BA cos θ: change any of the three

What changes	Induced emf
B	e = NA cos θ ΔBΔt
A	e = NB cos θ ΔAΔt
θ (turned)	e = NBA(cos θ₁ − cos θ₂)Δt

Turned from ⊥ field through 90°: Δφ = NBA.
Turned from ⊥ field through 180°: Δφ = 2NBA.
Removed from the field: Δφ = NBA.

FLUX GIVEN AS A FORMULA

12.1.5 φ as a function of time

Differentiate, then put in t.
Example: φ = at² + bt + c gives |e| = 2at + b.
"In the 5th second": from t = 4 to t = 5, use φ(5) − φ(4).
In a φ–t graph, the slope gives e.

e = slope of the φ–t graph

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Corner

Expanding or shrinking loop

e = B dAdt = B(2πr) drdt

The field stays the same; the area changes.
Elastic loop whose radius shrinks at rate drdt: use r at that instant.
Circle reshaped into a square of the same wire: area changes, so an emf appears.

Practise PYQs on this topicMagnetic Flux & Faraday's Law · 89 questions with solutions

CONCEPT

Induced Current, Charge & Heat

In one line
Once the circuit is closed, the emf drives a current. The total charge depends only on the change of flux, not on how fast it happens.

INDUCED CURRENT

12.2.1 Induced current

i = eR = −NR dφdt

R is the total resistance of the closed circuit.
Coil R with a galvanometer G in series: R_total = R + G.
Example: coil R and galvanometer 6R give i = N(φ₂ − φ₁)7Rt.
No closed path (a cut ring): emf yes, current no.

INDUCED CHARGE

12.2.2 Induced charge

q = NΔφR

q does not depend on the time taken.
A slower or faster change gives the same q.
Removed from the field: q = NBAR.
Turned through 180°: q = 2NBAR.
In an i–t graph, the area gives q; then Δφ = qR.

Faster magnet ⇒ e and i larger.
Same start and end ⇒ q the same.
So for two speeds: e₂ > e₁, i₂ > i₁, q₂ = q₁.

Area under i–t = q

POWER AND HEAT

12.2.3 Power and heat

P = i²R = e²R = N²R(dφdt)²

Quantity	Depends on time?	Depends on R?
e	yes	no
i	yes	yes
q	no	yes
P	yes	yes

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Coil of wire: power in terms of N and wire radius

P = e²R ∝ N a²

e ∝ N and R ∝ Na², where a is the radius of the wire.
Halving N and doubling a makes P twice as large.

Total heat when B falls with time

H = ∫ e²R dt

Use e(t) from the given φ(t), then integrate from start to the time flux becomes zero.

Practise PYQs on this topicMagnetic Flux & Faraday's Law · 89 questions with solutions

CONCEPT

Lenz’s Law & Eddy Currents

In one line
The induced current always flows so as to oppose the cause that produces it. This is energy conservation at work.

LENZ’S LAW

12.3.1 Direction of the induced current

Find the direction of the flux through the loop.
Is it increasing or decreasing?
Increasing: induced field is opposite. Decreasing: induced field is along it.
Use the right-hand rule to find the current.
Lenz’s law follows from conservation of energy.

Magnet	Current (seen from the magnet)	Near face	Force
N approaching	anticlockwise	N	repulsive
N receding	clockwise	S	attractive
S approaching	clockwise	S	repulsive
S receding	anticlockwise	N	attractive

RELATIVE MOTION

12.3.2 Only relative speed matters

Coil and magnet with the same velocity: no flux change, e = 0.
Coil and magnet moving apart, each at V: relative speed 2V, so e doubles.
A free ring moves in the direction of the magnet’s motion, to reduce relative motion.

LOOPS AND WIRES

12.3.3 Loops and wires: two common cases

Current in the outer of two concentric loops rising: inner current is opposite.
Two coils with current in the same sense pulled apart: current increases in both.
Ring approaching a straight wire: flux grows, so its field opposes the wire’s field there.

FALLING MAGNET

12.3.4 Magnet falling through a ring or pipe

Closed ring: a < g, both above and below the ring.
Cut ring: no current, a = g.
Long metal pipe: eddy currents; the magnet soon falls at almost constant speed.
After leaving the pipe: a = g.

Induced current opposes the fall

EDDY CURRENTS

12.3.5 Eddy currents

Slots raise the resistance of the eddy paths

Circulating currents in a thick metal in a changing flux (Foucault currents).
They cause heat loss.
Reduced by lamination and slotting.
Used in: electromagnetic damping (dead-beat galvanometer), electric brakes, induction furnace, speedometer, energy meter.

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Induced electric field

Closed circular lines; E rises to r = a, then falls

∮ E · dl = −dφdt

A changing B always produces an electric field in the space around it.
Its lines are closed loops: the field is non-conservative.
Work done on a charge q in one full round = qe, not zero.

Field inside and outside a circular region

r ≤ a: E = r2 dBdt

r ≥ a: E = a²2r dBdt

The emf in a loop that encloses the field depends only on the area that has the field: e = πa² dBdt.
A loop that does not enclose the field has e = 0.

Practise PYQs on this topicLenz's Law, Eddy Currents & Induced Electric Field · 36 questions with solutions

CONCEPT

Motional EMF: Straight Conductor

In one line
A conductor moving across a magnetic field cuts field lines. The magnetic force on its free charges builds up an emf.

ROD IN A FIELD

12.4.1 How the emf arises

Electrons feel F = q(v × B) and collect at one end.
The other end becomes positive.
Charges stop moving when qE = qvB.

e = Bvl

Rod moving right, B into the page

GENERAL FORM

12.4.2 Only perpendicular parts count

e = (v × B) · l

e = Bvl sin θ

Left: v at θ to B. Right: rod at θ to v.

If any two of B, v, l are parallel, e = 0.
θ is the angle between v and B, with l ⊥ both.
If B is given as H: B = μ₀H.
Solid cube: same formula across the faces perpendicular to v × B.

MOVING LOOP

12.4.3 Moving loop: which side cuts the field

Whole loop inside a uniform field: flux constant, e = 0.
Partly outside: only the side inside the field cuts lines.
Velocity ⊥ shorter side ⇒ use the shorter side as l.
Circular loop: e changes as it leaves, because the chord changes.

curved conductor: l = straight distance between its ends

Semicircle of radius r moving ⊥ its diameter: e = B(2r)v.

EARTH’S FIELD

12.4.4 Aeroplane, train, falling wire

Moving object	Field that counts	Emf
aeroplane (wings)	B_V = B sin δ	B_V v l
train (axle)	B_V	B_V v l
wire falling, east-west	B_H = B cos δ	B_H v l
vertical aerial, moving east	B_H	B_H v l

δ is the angle of dip.
At the magnetic equator B_V = 0, so a horizontal rod has e = 0.

FALLING WIRE

12.4.5 Wire dropped through height h

v = √2gh ⇒ e = B_H l √2gh

Keep the wire east-west: then it cuts B_H.
e keeps growing with time, since v grows.

Practise PYQs on this topicMotional EMF · 59 questions with solutions

CONCEPT

Force, Power & Terminal Velocity

In one line
A rod sliding on rails sweeps out area. The magnetic force opposes it, so an external force must do work, and that work appears as heat.

SLIDING ROD

12.5.1 Sliding rod: area swept out

In time t, the rod moves vt.
Area added: lvt. Flux added: Blvt.

e = dφdt = Blv

i = BlvR

F opposes the motion of the rod

FORCE

12.5.2 Force to keep v constant

F = Bil = B²l²vR

Magnetic force acts opposite to v.
At constant speed, applied force = magnetic force.

POWER

12.5.3 Power: work becomes heat

P_mech = Fv = B²l²v²R

P_heat = i²R = B²l²v²R

The two are equal: energy conservation.
Work to pull a loop out in time t:

W = (BA)²Rt (for a loop of area A)

PARALLEL RESISTORS

12.5.4 Rod between two resistors

The loop losing area and the loop gaining area carry opposite senses of current.
Rod pulled left, B into the page: left loop clockwise, right loop anticlockwise.
Find the rod’s current as for a cell of emf Blv.

TERMINAL VELOCITY

12.5.5 Terminal velocity

Rod falls: v rises, so i and F_m rise.
When F_m = mg, acceleration is zero.

v_T = mgRB²l²

F_m balances mg at v_T

The same result holds for a loop falling partly out of a field.
Use l between the rails, even if the rod is longer.

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Rod sliding down smooth inclined rails

Forces on the rod on the incline

B vertical: v_T = mgR sin θB²l² cos²θ

B normal to rails: v_T = mgR sin θB²l²

e = Bvl cos θ when B is vertical.
Balance the forces along the incline.

Sliding rod with a resistor in the rod and in each end

Treat the rod as a cell of emf Blv and internal resistance r.
Then solve it as an ordinary circuit.
Two equal end resistors R and rod resistance R: I = 2Blv3R, each end carries Blv3R.

Loop near a long straight wire

B = μ₀I2πx changes with x, so near and far arms have different emfs.
Net emf = difference of the two arm emfs.
Rod from x₁ to x₂ moving parallel to the wire: e = μ₀Iv2π ln(x₂x₁).

Practise PYQs on this topicMotional EMF · 59 questions with solutions

CONCEPT

Rotating Rod, Wheel & Disc

In one line
A rod turning in a field cuts lines at a speed that grows along its length, so use the average speed of ωl2.

ROTATING ROD

12.6.1 Rod with pivot at one end

e = 12 Bωl²

e = Bπl²n

Field ⊥ plane of rotation.
n = revolutions per second, ω = 2πn.
Average speed = ωl2.
Area swept per second = ½ωl².

Rod turning about one end O

Rod in a vertical plane, B_H ⊥ the plane: use B = B_H.
Fan blades in a horizontal plane: use B_V.

PIVOT IN BETWEEN

12.6.2 Pivot between the ends

e = 12 Bω(l₁² − l₂²) (l₁ > l₂)

Both ends are at a lower potential than the pivot (or both higher).
So the two arm emfs subtract.
Pivot at the centre: l₁ = l₂ ⇒ e = 0 between the ends.
Rod not reaching the pivot, from a to b:

e = 12 Bω(b² − a²)

WHEEL

12.6.3 Wheel with N spokes

e = 12 BωR²

Each spoke is an equal cell.
All spokes are in parallel.
e does not depend on the number of spokes.

Spokes are identical cells in parallel

DISC

12.6.4 Disc: Faraday generator

e = 12 BωR² = BπR²n

Disc turning in a field along its axis

A disc is many radial rods in parallel.
e is between the axis and the rim.
Same formula as the wheel.

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Semicircular loop entering a field

Field exists on one side of the axis only

e = 12 Bωr²

Area in the field grows at a constant rate.
e is constant in size and reverses each half turn.

B varying along the rod

e = ∫ B(x) ωx dx

Use this when B depends on distance from the pivot.

Practise PYQs on this topicRotating Conductors & AC Generator · 35 questions with solutions

CONCEPT

Rotating Coil & AC Generator

In one line
A coil turning in a uniform field has a flux that varies as a cosine, so its emf varies as a sine. That is alternating emf.

ROTATING COIL

12.7.1 Rotating coil: periodic EMI

φ = NBA cos ωt

e = NBAω sin ωt = e₀ sin ωt

e₀ = NBAω = NBA(2πf)

i = eR = i₀ sin ωt, i₀ = NBAωR

φ and e differ in phase by π2

POSITION OF THE COIL

12.7.2 Flux and emf at each position

Plane of coil	φ	e
⊥ field (ωt = 0)	maximum, NBA	zero
∥ field (ωt = 90°)	zero	maximum, NBAω
⊥ field again (ωt = 180°)	maximum, −NBA	zero

Phase difference between φ and e: π2.
e changes direction twice in each revolution.
From ⊥ position, e is largest at t = T4, zero at t = T2.

AVERAGE POWER

12.7.3 Average power over a cycle

P = e₀²2R = N²B²A²ω²2R

Average of sin²ωt over a cycle is 12.
Any shape of loop: use its area in A.
Semicircle: A = πr²2; ellipse: A = πab.

AC GENERATOR

12.7.4 AC generator

Coil rotating between the poles

Works on electromagnetic induction.
Armature: coil of many turns on a soft iron core.
Field magnet: strong magnet with N and S poles.
Slip rings: turn with the coil.
Brushes: fixed carbon contacts, connected to the load.
Dc generator: a commutator takes the place of slip rings.

Practise PYQs on this topicRotating Conductors & AC Generator · 35 questions with solutions

CONCEPT

Self-Inductance

In one line
A changing current in a coil changes its own flux, so the coil induces an emf in itself, opposing the change. L measures this.

DEFINITION

12.8.1 Definition of L

Nφ = Li ⇒ L = Nφi

e = −L didt

L is the flux linkage at unit current.
L is also the emf at unit didt.
L does not act while the current is steady.
Inductance is to current what inertia (mass) is to velocity.

Induced emf opposes a rise and supports a fall

UNIT

12.8.2 Unit: henry

1 H = 1 V sA = 1 WbA = 1 Ω s.
Dimension: [ML²T⁻²A⁻²].
In a φ–i graph, slope = L: the steepest line has the largest L.
In an i–t graph with constant slope, e is constant; rising i gives negative e.

Current reversed: Δi = i₁ + i₂.
Example: 5 A to −5 A is a change of 10 A.

WHAT L DEPENDS ON

12.8.3 What L depends on

Depends on N, area, length and μ of the core.
Does not depend on the current or its change.
Iron core: L = μ_rL₀.

SOLENOID

12.8.4 Solenoid: the most asked formula

L = μ₀μ_rN²Al = μ₀μ_rn²Al

Change	L becomes
N → 3N (same l)	9L
n → 2n (same l)	4L
all lengths × 2 (same n)	8L
N and μ_r changed	L ∝ μ_rN²

At a fixed N: L ∝ Al.
Per unit length: Ll = μ₀n²A = μ₀π(nd2)².
Near the end of a solenoid, L is half of its value at the middle.

WIRE LENGTH

12.8.5 Solenoid from a wire of length x

x = √4πlLμ₀

x = N(2πr), so N²πr² = x²4π.
With R, m, density D and resistivity ρ: x² = RmρD.

OTHER COILS

12.8.6 Formulae for L

Coil	L
circular, radius r	μ₀N²πr2
solenoid	μ₀N²Al
toroid, radius r	μ₀N²A2πr
square, side a	2√2 μ₀N²aπ

Practise PYQs on this topicSelf-Inductance & Energy · 91 questions with solutions

CONCEPT

Energy Stored in an Inductor

In one line
Work is done against the back emf while the current builds up. It is stored in the magnetic field of the coil.

ENERGY

12.9.1 Energy stored

U = 12Li²

Stored in the magnetic field, not in the coil’s wire.
U ∝ i²: half the current, one-fourth the energy.
At i = 1 A, L = 2U.

U = 12Nφi

Comes from L = Nφi.
Use it when flux and turns are given.

RATE OF STORING

12.9.2 Power given to a coil

P = dUdt = Li didt

Two coils with same P and same didt: Li is the same.
So i ∝ 1L and U ∝ 1L.
Hence U₁U₂ = L₂L₁.

CHANGE IN ENERGY

12.9.3 Change in energy

ΔU = 12L(i₂² − i₁²)

Find L from e first: L = edidt.
Same E and R, different L: U ∝ L.

ENERGY DENSITY

12.9.4 Energy density

u = B²2μ₀μ_r

U = B²2μ × Al

COMBINATIONS

12.9.5 Energy in a group of inductors

Find L_eq, then U = 12L_eqI².
Three L in series: U = 32LI².
Two equal L in parallel: L_eq = L2.

Practise PYQs on this topicSelf-Inductance & Energy · 91 questions with solutions

CONCEPT

Mutual Inductance

In one line
A changing current in one coil changes the flux through a neighbouring coil and induces an emf in it.

DEFINITION

12.10.1 Definition of M

N₂φ₂ = Mi₁

e₂ = −M di₁dt

Flux of the primary links the secondary

M is the same both ways: M₁₂ = M₂₁.
So φ_P from i_Q and φ_Q from i_P both use one M.
Unit: henry; same dimension as L.
If I = I₀ sin ωt, then e₀ = MI₀ω.
Flux per turn given: M = N₂φ₂i₁.

DEPENDENCE

12.10.2 What M depends on

Turns N₁, N₂ and the area of the coils.
Medium: μ_r of the core.
Distance: larger d, smaller M.
Relative position and orientation: at 90°, M = 0.
Not on the currents or their rate of change.

COUPLING

12.10.3 Coefficient of coupling k

M = k√L₁L₂, 0 ≤ k ≤ 1

Coils linked fully, partly or not at all

All flux linked: k = 1, M = √L₁L₂.
k = M√L₁L₂.

CONCENTRIC LOOPS

12.10.4 Concentric loops (R ≫ r)

M = μ₀πr²2R

B of the big loop at the centre: μ₀I2R.
Flux through the small loop: Bπr².
M ∝ r²R.
With N₁, N₂ turns: M = μ₀N₁N₂πr²2R.

Small loop at the centre of a big loop

COAXIAL SOLENOIDS

12.10.5 Coaxial solenoids

M = μ₀n₁n₂πr₁²l = μ₀N₁N₂Al

Flux exists only over the inner area, πr₁².
ML₁ = n₂n₁.

TRANSFORMER

12.10.6 Transformer

Same flux per turn in both coils

V_SV_P = N_SN_P = i_Pi_S

Works on ac only: it needs a changing flux.
Frequency does not change.
Volts per turn are the same in both coils.
Step-up: N_S > N_P; step-down: N_S < N_P.
Efficiency η = P_outP_in.
Losses: copper, eddy (laminated core), hysteresis, flux leakage.

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Square loops

Small loop at the centre of a large square

small square a inside large square b: M = μ₀4π 8√2 a²b

small circle r inside large square L: M = 2√2 μ₀r²L

Field at the centre of a square of side L: four finite wires at distance L2.

Small coil inside a long solenoid

M = μ₀ n N A

n = turns per unit length of the solenoid.
N, A = turns and area of the small coil (A must be inside the solenoid).
Current in a ring outside the solenoid: use the solenoid’s area, not the ring’s.

Practise PYQs on this topicMutual Inductance · 64 questions with solutions

CONCEPT

Combination of Inductors

In one line
Without mutual coupling, inductors combine like resistors: add in series, add reciprocals in parallel.

SERIES

12.11.1 Series: same current

far apart: L = L₁ + L₂

close, fields aiding: L = L₁ + L₂ + 2M

close, fields opposing: L = L₁ + L₂ − 2M

Two identical coils (M = L) with opposite windings: L_eq = 0.
This is how resistance boxes avoid inductance: the wire is doubled back.

PARALLEL

12.11.2 Parallel: same voltage

far apart: 1L = 1L₁ + 1L₂

close, fields aiding: L = L₁L₂ − M²L₁ + L₂ − 2M

Current divides as I₁L₁ = I₂L₂, so I₂I₁ = L₁L₂.
A coil cut in two equal parts, joined in parallel: L4.
Minimum L: all in parallel. Maximum L: all in series.

TRICKY WIRING

12.11.3 Tricky wiring: trace the nodes

Give the same letter to points joined by a plain wire.
Redraw between the two end letters.
Often all inductors turn out to be directly between A and B.

VOLTAGE ACROSS AN INDUCTOR

12.11.4 Voltage across an inductor

V_L = L didt

Moving along the current	Change in potential
through R	−iR
through L	−L didt
cell, + to −	−E
cell, − to +	+E

If i is decreasing, didt is negative: put the sign in the formula.
Series R and L: V_AB = iR + L didt.
Measure V at the same i for rising and falling current: add to get R, subtract to get L.

Practise PYQs on this topicCombination of Inductors, LR & LC Circuits · 35 questions with solutions

CONCEPT

LR & LC Circuits

In one line
An inductor does not let a current change suddenly. In LR circuits current changes exponentially, and in LC circuits it oscillates.

LR CIRCUIT

12.12.1 LR circuit: growth and decay

τ = LR

i₀ = ER

Growth: i = 0.63 i₀ at t = τ

Decay: i = 0.37 i₀ at t = τ

τ is the time constant, in seconds.
Half value of i₀ at t = 0.693 τ.
Steady state: L acts as a plain wire, i = ER.
At the instant of closing: L acts as an open circuit.
At sudden opening, a large emf causes a spark; a capacitor across the key prevents it.

TWO BULBS

12.12.2 Two bulbs: when a switch opens

The inductor keeps its current going through any closed path.
If both bulbs lie on that path, both fade together.

LC OSCILLATION

12.12.3 LC oscillation

Charged C discharging through L

ω = 1√LC, f = 12π√LC

q₀²2C = 12LI₀² ⇒ I₀ = q₀√LC = V√CL

No resistance ⇒ no energy loss.
At t = 0: all energy is electric, in C.
Current reverses each half cycle.
To stop a spark: a capacitor must hold the inductor’s energy.

12LI² = 12CV² ⇒ C = L(IV)²

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Corner

Using the equations

Growth: i = i₀(1 − e^−t/τ)

Decay: i = i₀e^−t/τ

didt at t = 0 is EL

Charge in time τ during growth: q = ELeR².
Energy reaches half its maximum at i = i₀√2.
Heat rate equals energy-storing rate at i = i₀2: t = τ ln 2.

Circuits with several branches

At t = 0: each inductor is an open circuit.
At t = ∞: each inductor is a plain wire.
Just after a switch opens, an inductor keeps its current for that instant.

Inductor as a part of a circuit

V_L = E − iR

The ratio of voltages at two currents comes directly from this.

Practise PYQs on this topicCombination of Inductors, LR & LC Circuits · 35 questions with solutions

EXAM-DAY CARD

Just see this before the exam

e = −Ndφdt

Faraday

q = NΔφR

charge

e = Bvl

motional

e = 12Bωl²

rotating rod

e₀ = NBAω

rotating coil

U = 12Li²

energy

FORMULA SHEET

Whole chapter on one page

Flux & Faraday

φ = BA cos θ

e = −Ndφdt

i = eR

q = NΔφR

Motional

e = Bvl sin θ

F = B²l²vR

P = B²l²v²R

v_T = mgRB²l²

Rotating

rod, disc: e = 12Bωl²

e₀ = NBAω

P = e₀²2R

φ, e differ by π2

Self-inductance

L = Nφi

e = −Ldidt

L = μ₀N²Al

U = 12Li²

Mutual

M = N₂φ₂i₁

e₂ = −Mdi₁dt

M = k√L₁L₂

concentric: M = μ₀πr²2R

Circuits

series: L₁ + L₂ ± 2M

parallel: 1L = 1L₁ + 1L₂

τ = LR, i = i₀(1 − e^−t/τ)

f = 12π√LC

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

180° turn: Δφ = NBA

Whole loop inside field: e = Bvl

More spokes, more emf

Series with galvanometer: use R only

CORRECT ✓

180° turn: Δφ = 2NBA

Flux constant: e = 0

e = ½BωR², whatever the spokes

Use R + G in i and q