Electromagnetic induction and AC for NEET

Electromagnetic induction and alternating current are two NCERT chapters that belong together. Induction explains how a changing magnetic flux makes a voltage; AC is what you get when you spin a coil in a field and use that voltage. NEET questions from these chapters are usually direct: an induced EMF, the impedance of an LCR circuit, the resonant frequency, the power factor, or a transformer's turns ratio.

The conceptual traps are the phase relations in AC circuits and the direction of an induced current. Both become easy once you tie them to a picture.

Magnetic flux and Faraday's law

The magnetic flux through a flat area A in a uniform field B is Φ = BA cos θ, where θ is the angle between B and the normal to the area. Its unit is the weber (Wb).

Faraday's law: the EMF induced in a coil of N turns equals the rate of change of flux linkage:

ε = −N dΦ/dt

Flux can change in three ways: change B, change the area, or change the angle (rotate the coil).

The charge that flows through a circuit of resistance R when the flux changes by ΔΦ is q = NΔΦ/R. It depends only on the total change, not on how fast it happened.

Lenz's law

The minus sign is Lenz's law: the induced current flows in the direction that opposes the change in flux that caused it.

• A magnet's north pole approaching a coil makes the near face of the coil a north pole, repelling it.
• The same north pole moving away makes that face a south pole, attracting it.

Lenz's law is a consequence of conservation of energy. If the induced current helped the change, you would get electrical energy for nothing.

Motional EMF

A rod of length l moving with velocity v perpendicular to a field B has an EMF ε = Blv across its ends. You can see it either as the Lorentz force pushing charges along the rod, or as the flux change through the circuit the rod completes.

• If the rod slides on rails closing a circuit of resistance R, the current is Blv/R.
• The magnetic force on the rod is B²l²v/R, opposing the motion (Lenz again). To keep the speed constant, you must apply an equal force, and the power you supply, B²l²v²/R, appears as heat in R.
• A rod of length l rotating about one end with angular speed ω in a perpendicular field: ε = ½Bωl².

Eddy currents

A changing flux through a bulk metal plate induces swirling eddy currents in it. They oppose the motion that causes them.

• Uses: electromagnetic damping in galvanometers, magnetic braking in some trains, induction furnaces, and electric power meters.
• Losses: they heat transformer and motor cores. Laminating the core (thin insulated sheets) cuts the eddy current paths and reduces this loss.

Self and mutual inductance

Self-inductance: a coil's own flux linkage is proportional to its current, NΦ = LI. A changing current induces a back EMF ε = −L dI/dt, opposing the change. The unit is the henry (H).

• For a long solenoid of N turns, length l and area A: L = μ₀N²A/l = μ₀n²Al.
• Energy stored: U = ½LI². This is the magnetic twin of ½CV².

Mutual inductance: a changing current in one coil induces an EMF in a nearby coil: ε₂ = −M dI₁/dt. For two long coaxial solenoids, M = μ₀n₁n₂Al, where A and l belong to the inner one.

AC generator and rms values

A coil of N turns and area A spinning at angular speed ω in a field B produces

ε = NBAω sin ωt, with peak value ε₀ = NBAω.

For a sinusoidal current I = I₀ sin ωt:

• rms value: I_rms = I₀/√2 ≈ 0.707 I₀. The same for voltage. The 230 V of Indian mains is an rms value.
• The average over a full cycle is zero; over a half cycle, it is 2I₀/π.

AC meters read rms values.

AC through R, L and C

X_L grows with frequency; X_C falls with frequency. For DC (ω = 0), an inductor is a plain wire and a capacitor is an open circuit.

Series LCR circuit and resonance

With R, L and C in series, the voltages across them are out of phase, so they add as phasors, not as numbers:

• Impedance: Z = √[R² + (X_L − X_C)²]
• Phase angle: tan φ = (X_L − X_C)/R. If X_L > X_C, the circuit is inductive and current lags; if X_C > X_L, it is capacitive and current leads.

Resonance happens when X_L = X_C:

• ω₀ = 1/√(LC), or f₀ = 1/(2π√(LC)).
• Z = R, its minimum, so the current is maximum.
• Current and voltage are in phase, and the power factor is 1.
• The voltages across L and C are equal and opposite, and each can be larger than the supply voltage.

Power in AC circuits

P = V_rms I_rms cos φ, where the power factor cos φ = R/Z.

• Pure resistor: cos φ = 1, full power.
• Pure inductor or capacitor: cos φ = 0, no average power. The current still flows; it is called wattless current. In general, the wattless component is I_rms sin φ.

Transformers

A transformer changes AC voltage using mutual induction between two coils on a shared iron core. For an ideal transformer:

V_s/V_p = N_s/N_p = I_p/I_s

• Step-up (N_s > N_p) raises voltage and lowers current. Step-down does the reverse.
• Power in = power out for an ideal transformer.
• Real losses: copper (I²R heating of windings), eddy currents (reduced by laminating), hysteresis (reduced by a soft iron core) and flux leakage.

Power is transmitted at high voltage so the current, and therefore the I²R loss in the lines, is small.

Worked numericals

Example 1: a rod on rails

A 1 m rod slides at 4 m s⁻¹ on rails in a 0.5 T field. The circuit resistance is 4 Ω.

• ε = Blv = 0.5 × 1 × 4 = 2 V.
• I = 2/4 = 0.5 A.
• Force needed to keep it moving: F = BIl = 0.5 × 0.5 × 1 = 0.25 N.
• Power: Fv = 0.25 × 4 = 1 W, which equals εI = 2 × 0.5 = 1 W ✓.

Example 2: EMF and charge in a coil

A coil of 100 turns and area 0.02 m² is in a 0.5 T field normal to it. The field falls to zero in 0.1 s. The coil's resistance is 5 Ω.

• ΔΦ per turn = 0.5 × 0.02 = 0.01 Wb.
• ε = NΔΦ/Δt = 100 × 0.01/0.1 = 10 V.
• Charge: q = NΔΦ/R = 100 × 0.01/5 = 0.2 C. Check: I = 10/5 = 2 A for 0.1 s gives 0.2 C ✓.

Example 3: series LCR circuit

A series circuit has R = 30 Ω, X_L = 80 Ω and X_C = 40 Ω, across 200 V rms.

• Z = √(30² + 40²) = √2,500 = 50 Ω.
• I_rms = 200/50 = 4 A.
• Power factor = 30/50 = 0.6; the circuit is inductive, so current lags.
• Power = 200 × 4 × 0.6 = 480 W. Check: I²R = 16 × 30 = 480 W ✓.

Example 4: resonant frequency

L = 0.1 H and C = 10 μF.

• LC = 0.1 × 10 × 10⁻⁶ = 10⁻⁶, so √(LC) = 10⁻³.
• ω₀ = 1/10⁻³ = 1,000 rad s⁻¹, and f₀ = 1,000/(2π) ≈ 159 Hz.

Example 5: transformer

A transformer steps 220 V down to 11 V. The primary has 1,000 turns, and the secondary supplies 10 A.

• N_s = 1,000 × 11/220 = 50 turns.
• For an ideal transformer, I_p = 10 × 11/220 = 0.5 A. Power: 220 × 0.5 = 11 × 10 = 110 W on both sides ✓.

Practice MCQs

1. The peak value of a 220 V rms supply is about: (a) 156 V (b) 220 V (c) 311 V (d) 440 V
2. At resonance, the impedance of a series LCR circuit equals: (a) zero (b) R (c) X_L + X_C (d) √(R² + X_L²)
3. If the AC frequency is doubled, X_L and X_C become: (a) both doubled (b) doubled and halved (c) halved and doubled (d) both halved
4. The power factor of a circuit with a pure inductor is: (a) 0 (b) 0.5 (c) 0.707 (d) 1
5. The energy stored in a 2 H inductor carrying 3 A is: (a) 3 J (b) 6 J (c) 9 J (d) 18 J
6. Lenz's law is a consequence of the conservation of: (a) charge (b) momentum (c) energy (d) mass
7. The number of turns of a solenoid is doubled, keeping its length and area the same. Its self-inductance becomes: (a) 2 times (b) 4 times (c) half (d) unchanged
8. The reactance of a 100 μF capacitor at 50 Hz is about: (a) 3.2 Ω (b) 31.8 Ω (c) 318 Ω (d) 0.03 Ω

Answers

1. (c) 220 × √2 ≈ 311 V.
2. (b) X_L = X_C, so Z = R.
3. (b) X_L ∝ f and X_C ∝ 1/f.
4. (a) φ = 90°, cos φ = 0.
5. (c) ½ × 2 × 3² = 9 J.
6. (c) An induced current that aided the change would create energy from nothing.
7. (b) L ∝ N².
8. (b) X_C = 1/(2π × 50 × 10⁻⁴) = 1/0.0314 ≈ 31.8 Ω.

What to do next

• Practise Lenz's law with five magnet-and-coil pictures, marking the pole on the near face each time.
• Write the R, L, C table (reactance, phase, power) from memory.
• Solve five LCR problems: find Z, I, power factor and power, and check P = I²R each time.
• Revise moving charges and magnetism if magnetic force and flux still feel unfamiliar.

Next in the syllabus: electromagnetic waves, then ray optics for NEET.