Electromagnetic Induction notes: ISC Class 12 Physics

1. Magnetic flux and Faraday’s laws

Magnetic flux through a flat coil of area A in a uniform field: $\Phi = \vec B \cdot \vec A = BA\cos\theta$ , θ between $\vec B$ and the normal. Unit: weber, 1 Wb = 1 T m². Flux linkage of N turns: $N\Phi$ .
If the angle is given from the plane of the coil, use θ = 90° minus that angle.
First law: an emf is induced whenever the flux linked with a circuit changes, and only while it changes. Second law: $\varepsilon = -N\,\dfrac{d\Phi}{dt}$ (the minus sign is Lenz’s law).
The flux can change through B, A or θ. On a Φ–t graph the emf per turn is the (negative) slope: a steady flux, however large, gives no emf.
Charge that flows round a closed coil of resistance R: $q = \dfrac{N\Delta\Phi}{R}$ , independent of how fast the change happens.

rising: slope 2.0 mWb s⁻¹
steady: slope zero
falling: slope −6.0 mWb s⁻¹

The emf per turn is the slope: 2.0 mV, then zero, then 6.0 mV in the opposite sense.

Formulas

Flux
$\Phi = BA\cos\theta$
Faraday’s law; charge
$\varepsilon = -N\,\dfrac{d\Phi}{dt},\qquad q = \dfrac{N\Delta\Phi}{R}$

Key terms

Magnetic flux: BA cos θ for a flat area A in a uniform field, θ measured from the normal; unit weber.
Flux linkage: Total flux through all the turns of a coil, NΦ.

2. Lenz’s law and motional emf

Lenz’s law: the induced current flows so as to oppose the change of flux that produces it (not the flux itself). Flux increasing: induced field against B; flux decreasing: induced field along B.
Steps: (1) is the flux through the loop increasing or decreasing, and which way does it point? (2) choose the induced field that opposes the change; (3) right-hand rule for the current.
Lenz’s law follows from conservation of energy: because the induced current opposes the motion, work must be done, and that work becomes the electrical energy.
Motional emf: a bar of length l moving at v on rails in a perpendicular field: $\Phi = Blx$ , so $\varepsilon = Bl\,\dfrac{dx}{dt} = Blv$ . Same result from forces: $qvB$ on each free charge balances $qE$ , so $\varepsilon = El = Blv$ .
With circuit resistance R: $I = Blv/R$ ; opposing force $F = BIl = B^2l^2v/R$ ; power to keep v steady $P = Fv = (Blv)^2/R = I^2R$ .

P = (Blv)²/R for B = 0.40 T, l = 0.50 m, R = 0.80 Ω

P ∝ v²: the emf, the current and the drag force each grow in proportion to v.

Formulas

Motional emf; drag force
$\varepsilon = Blv,\qquad F = \dfrac{B^2l^2v}{R}$
Power
$P = \dfrac{(Blv)^2}{R}$

Key terms

Lenz’s law: The induced current opposes the change of flux that causes it.
Motional emf: The emf Blv across a conductor moving through a magnetic field.

3. Eddy currents (qualitative)

Eddy currents are induced currents circulating in closed loops inside a bulk conductor when the flux through it changes (the metal moves in a field or the field changes).
Effects: heating of the metal; a force that opposes the relative motion (Lenz’s law).
Uses: induction furnaces and cooktops (heating); electromagnetic brakes on trains; dead-beat galvanometers (damping by the metal frame); speedometers and energy-meter discs.
Unwanted: heating of transformer and motor cores. Lamination (thin sheets insulated by varnish) confines the eddy currents to small, high-resistance paths and reduces the loss. Slots cut in a plate do the same.
ISC treats eddy currents qualitatively only.

Key terms

Eddy currents: Induced currents that circulate inside a bulk piece of metal when the flux through it changes.
Lamination: Building a core from thin sheets insulated from one another to reduce eddy currents.

4. Self-induction

The flux linkage of a coil is proportional to its own current: $N\Phi = LI$ , and $\varepsilon = -L\,\dfrac{dI}{dt}$ . Self-inductance L, unit henry (1 H = 1 Wb A⁻¹ = 1 V s A⁻¹).
Long solenoid (N turns, length l, area A, n = N/l): $B = \mu_0NI/l$ , $N\Phi = \mu_0N^2AI/l$ , so $L = \dfrac{\mu_0N^2A}{l} = \mu_0n^2Al$ ; with a core $\mu_r$ times larger.
L depends on the turns, size, shape and core, not on the current.
Back emf opposes the change of current: the current grows gradually when switched on; a large emf (spark) appears when it is switched off. It depends on dI/dt, so it can be large when I is zero. Inductance is “electrical inertia”.

Formulas

Self-induction
$N\Phi = LI,\qquad \varepsilon = -L\,\dfrac{dI}{dt}$
Long solenoid
$L = \dfrac{\mu_0N^2A}{l}$

Key terms

Self-inductance: Flux linkage of a coil per unit current through it, L = NΦ/I; unit henry.
Back emf: The emf −L dI/dt induced in a coil by a change of its own current.

5. Mutual induction

A current $I_1$ in coil 1 links flux with coil 2: $N_2\Phi_2 = MI_1$ and $\varepsilon_2 = -M\,\dfrac{dI_1}{dt}$ . Mutual inductance M, in henry; $M_{12} = M_{21}$ .
M grows with the turns and areas, with closer, coaxial placing and with a soft-iron core through both coils.
Coaxial solenoids (inner radius $r_1$ , $n_1$ and $n_2$ turns per metre, common length l): the outer field $\mu_0n_2I_2$ passes through the inner area $\pi r_1^2$ and links $n_1l$ turns, so $M = \mu_0n_1n_2\pi r_1^2l$ . Only the inner area counts.

Formulas

Mutual induction
$N_2\Phi_2 = MI_1,\qquad \varepsilon_2 = -M\,\dfrac{dI_1}{dt}$
Coaxial solenoids
$M = \mu_0n_1n_2\pi r_1^2l$

Key terms

Mutual inductance: Flux linkage of one coil per unit current in another, M = N₂Φ₂/I₁; unit henry.

6. The transformer and power transmission

Principle: mutual induction. Primary ( $N_p$ ) and secondary ( $N_s$ ) on a laminated soft-iron core; the alternating primary current sets up an alternating flux that links the secondary. It does not work on steady DC.
Ideal (no flux leakage, no losses): $\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} = \dfrac{I_p}{I_s}$ . Step-up: $N_s > N_p$ ; step-down: $N_s < N_p$ . It never increases power.
Efficiency $\eta = \dfrac{V_sI_s}{V_pI_p}$ (95–99% for large transformers). Losses: copper ( $I^2R$ ; thick wire), eddy currents (laminated core), hysteresis (soft iron or silicon steel), flux leakage (windings one over the other).
Transmission: line loss $I^2R = \dfrac{P^2R}{V^2}$ ; raising V k times cuts the loss $k^2$ times. Power is stepped up for long lines and stepped down to 230 V, 50 Hz near users.

loss = (P/V)²R for P = 500 kW, R = 5.0 Ω

Tripling the voltage from 11 kV to 33 kV cuts the loss nine times.

Formulas

Ideal transformer
$\dfrac{V_s}{V_p} = \dfrac{N_s}{N_p} = \dfrac{I_p}{I_s}$
Line loss
$P_{loss} = \dfrac{P^2R}{V^2}$

Key terms

Transformer: Device that changes an alternating voltage by mutual induction between two coils on a common core.
Efficiency of a transformer: Output power divided by input power.

Exam tips

Flux: is the angle given from the plane or from the normal?
Direction: change of flux first, then induced field, then current.
Motional emf: write Φ = Blx and differentiate; power (Blv)²/R.
Back emf depends on dI/dt; coaxial M uses the inner area; transformer V follows the turns ratio and I the inverse ratio.

1. Magnetic flux and Faraday’s laws

Formulas

Key terms

2. Lenz’s law and motional emf

Formulas

Key terms

3. Eddy currents (qualitative)

Key terms

4. Self-induction

Formulas

Key terms

5. Mutual induction

Formulas

Key terms

6. The transformer and power transmission

Formulas

Key terms

Exam tips

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