Electromagnetic Induction and Alternating Currents | NEET| JEE | CUET

Electromagnetic Induction and Alternating Currents

1. Introduction to Electromagnetic Induction

Electromagnetic Induction is the phenomenon of generation of an induced electromotive force (EMF) and hence induced current in a closed circuit due to a change in magnetic flux linked with the circuit. This discovery by Michael Faraday in 1831 established the symmetrical relationship between electricity and magnetism and laid the foundation for modern electrical technology, including generators, transformers, motors, and inductors.

Faraday’s experiments demonstrated that a changing magnetic field produces an electric field, which can drive charges and produce current.

2. Magnetic Flux

Magnetic flux (Φ_B) through a surface is defined as the surface integral of the magnetic field:

$$\Phi_B = \int \mathbf{B} \cdot d\mathbf{A}$$

For uniform B and plane surface of area A:

$$\Phi_B = B A \cos\theta$$

where θ is the angle between B and the normal to the surface.

• SI Unit: Weber (Wb) or Tesla·m².
• Flux is a scalar quantity and can be positive or negative depending on the chosen direction.

3. Faraday’s Laws of Electromagnetic Induction

First Law: Whenever the magnetic flux linked with a closed circuit changes, an induced EMF is produced in the circuit. The induced EMF lasts as long as the flux is changing.

Second Law (Quantitative): The magnitude of the induced EMF is equal to the rate of change of magnetic flux linked with the circuit:

$$\varepsilon = -\frac{d\Phi_B}{dt}$$

For a coil with N turns:

$$\varepsilon = -N \frac{d\Phi_B}{dt}$$

(The negative sign is incorporated in Lenz’s law.)

4. Lenz’s Law

Lenz’s law gives the direction of induced EMF and current. It states that the induced current will always flow in a direction such that it opposes the change in magnetic flux that produced it. This is a consequence of the law of conservation of energy.

Illustration:

• When a north pole of a magnet approaches a coil, the induced current creates a north pole on the facing side to repel the magnet (opposing the increase in flux).
• When the magnet is withdrawn, the induced current creates a south pole to attract the magnet (opposing the decrease in flux).

Lenz’s law ensures that mechanical work done against the opposing force is converted into electrical energy.

5. Motional EMF

When a conductor moves in a magnetic field, an EMF is induced due to the Lorentz force on free charges.

Rod moving perpendicular to uniform B:

A conducting rod of length l moving with velocity v perpendicular to B experiences motional EMF:

$$\varepsilon = B l v$$

Derivation: Charges experience force F = q v B. Separation of charges creates an electric field until electrostatic force balances magnetic force: qE = q v B ⇒ E = v B ⇒ ε = E l = B l v.

General Expression for Motional EMF:

$$\varepsilon = (\mathbf{v} \times \mathbf{B}) \cdot \mathbf{l}$$

EMF in a rotating rod or disc: For a rod of length l rotating with angular velocity ω about one end in perpendicular B:

$$\varepsilon = \frac{1}{2} B \omega l^2$$

6. Eddy Currents

When a bulk piece of conductor moves in a changing magnetic field or rotates, induced currents are produced within the body. These circulating currents are called eddy currents.

• They cause energy loss as heat (I²R loss).
• Applications: Electromagnetic damping (in galvanometers, induction brakes), induction furnaces (heating), speedometers.
• Minimization: Lamination of cores (thin insulated sheets reduce path for eddy currents).

7. Self-Induction and Inductance

Self-Induction: The phenomenon in which a changing current in a coil induces an EMF in the same coil due to the changing magnetic flux it produces.

The induced EMF opposes the change in current (Lenz’s law).

Self-Inductance (L):

$$\varepsilon = -L \frac{dI}{dt}$$

• SI Unit: Henry (H), where 1 H = 1 V·s/A.
• For a long solenoid:
  $$L = \mu_0 n^2 A l$$
  (n = turns per unit length, A = area, l = length).
• Inductance depends on geometry and medium (μ).

Energy Stored in an Inductor:

When current increases from 0 to I:

$$U = \frac{1}{2} L I^2$$

(This energy is stored in the magnetic field.)

Growth and Decay of Current in LR Circuit:

• Growth: I = I₀ (1 – e^{-t/τ}), where τ = L/R (time constant).
• Decay: I = I₀ e^{-t/τ}.

8. Mutual Induction

When changing current in one coil (primary) induces EMF in a neighbouring coil (secondary), the phenomenon is called mutual induction.

Mutual Inductance (M):

$$\varepsilon_2 = -M \frac{dI_1}{dt}$$

• For two coaxial solenoids or closely wound coils: M = μ₀ N₁ N₂ A / l.
• Coefficient of coupling k = M / √(L₁ L₂) (0 ≤ k ≤ 1).

Applications: Transformers, ignition coils, wireless charging.

9. Introduction to Alternating Currents

Alternating Current (AC) periodically reverses its direction. It is produced by AC generators (alternators) using the principle of electromagnetic induction.

Instantaneous Values:

• Voltage: v = v₀ sin(ωt) or v = v₀ sin(ωt + φ)
• Current: i = i₀ sin(ωt) or i = i₀ sin(ωt + φ)
• ω = 2πf (angular frequency), f = frequency (usually 50 Hz in India).

Peak, RMS, and Average Values:

• Peak (maximum) value: v₀ or I₀
• Root Mean Square (RMS) value (effective value):
  $$V_{\text{rms}} = \frac{v_0}{\sqrt{2}}, \quad I_{\text{rms}} = \frac{I_0}{\sqrt{2}}$$
  (RMS values are used for power calculations as they give the same heating effect as DC.)
• Average value over one cycle for sinusoidal AC: 0 (due to symmetry).

Phasor Representation: AC quantities are represented as rotating vectors (phasors) for easy addition of voltages and currents with phase differences.

10. AC Circuit Elements and Their Behaviour

1. Purely Resistive Circuit:
   • Current in phase with voltage (phase difference φ = 0).
   • Impedance Z = R.
   • Power: P = V_rms I_rms = I_rms² R (average power).
2. Purely Inductive Circuit (L):
   • Voltage leads current by 90° (or current lags voltage by π/2).
   • Inductive reactance: X_L = ωL = 2πf L.
   • Impedance Z = X_L.
   • Average power = 0 (no energy dissipation; energy oscillates between source and magnetic field).
3. Purely Capacitive Circuit (C):
   • Current leads voltage by 90° (or voltage lags current by π/2).
   • Capacitive reactance: X_C = 1/(ωC).
   • Impedance Z = X_C.
   • Average power = 0 (energy oscillates between source and electric field).

11. Series LCR Circuit (Resonant Circuit)

In a series combination of resistor R, inductor L, and capacitor C:

• Impedance:
  $$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
• Phase difference φ between voltage and current:
  $$\tan\phi = \frac{X_L - X_C}{R}$$
• Voltage across components: V_R = I R, V_L = I X_L, V_C = I X_C.

Resonance:

• Occurs when X_L = X_C ⇒ ω₀ = 1/√(LC) or f₀ = 1/(2π√(LC)).
• At resonance: Z = R (minimum), current is maximum (I_max = V/R), phase difference = 0.
• Quality factor (Q-factor): Q = ω₀ L / R = 1/(ω₀ C R). Higher Q means sharper resonance.
• Applications: Tuning circuits in radio/TV, filters, induction heating.

Power in AC Circuit:

Average power:

$$P_{\text{avg}} = V_{\text{rms}} I_{\text{rms}} \cos\phi$$

where cosφ is the power factor.

• Power factor = R/Z (ranges from 0 to 1).
• For pure L or C: cosφ = 0 (wattless current).

12. Transformers

A transformer transfers electrical energy from one circuit to another without change in frequency, using mutual induction.

Working Principle: Alternating voltage in primary produces changing flux, inducing EMF in secondary.

Ideal Transformer (no losses):

$$\frac{V_s}{V_p} = \frac{N_s}{N_p} = \frac{I_p}{I_s}$$

• Step-up: N_s > N_p (voltage increases, current decreases).
• Step-down: N_s < N_p.

Efficiency:

$$\eta = \frac{\text{Output Power}}{\text{Input Power}} \times 100\% = \frac{V_s I_s}{V_p I_p} \times 100\%$$

Energy Losses in Transformer:

• Copper losses (I²R in windings)
• Iron losses (hysteresis + eddy currents in core)
• Flux leakage, humming, etc.
• Minimized by using laminated soft iron core, proper winding, etc.

Uses: Power transmission (high voltage, low current reduces I²R loss), voltage regulation in appliances, isolation.

13. AC Generator (Alternator)

An AC generator converts mechanical energy into electrical energy based on electromagnetic induction.

Principle: A coil rotated in a uniform magnetic field experiences changing flux, inducing alternating EMF:

$$\varepsilon = N B A \omega \sin(\omega t)$$

Peak EMF: ε₀ = N B A ω.

Parts: Armature (coil), field magnet, slip rings, brushes.

14. Important Formula Summary

• Induced EMF: ε = – dΦ/dt or –N dΦ/dt
• Motional EMF: ε = B l v
• Self-Inductance: ε = –L dI/dt
• Energy in inductor: (1/2) L I²
• Inductive reactance: X_L = ωL
• Capacitive reactance: X_C = 1/(ωC)
• Impedance in LCR: Z = √[R² + (X_L – X_C)²]
• Resonance frequency: f₀ = 1/(2π√(LC))
• RMS values: V_rms = V₀/√2
• Average power: P = V_rms I_rms cosφ
• Transformer: V_s / V_p = N_s / N_p

15. Applications and Real-World Relevance

• Power Generation and Transmission: AC is preferred over DC for long-distance transmission due to easy voltage transformation.
• Household Appliances: Motors, fans, refrigerators, chargers use induction principles.
• Electromagnetic Braking and Damping: In trains and instruments.
• Wireless Power Transfer: Based on mutual induction.
• Resonant Circuits: In communication systems, medical imaging (MRI uses strong fields and resonance).

Common Misconceptions:

• Induced EMF does not depend on resistance of the circuit (only induced current does).
• In pure L or C circuits, average power is zero, but instantaneous power is not.
• Transformers work only with AC, not DC.

Examination Strategy:

• Master direction using Lenz’s law and Fleming’s right-hand rule (for generators).
• Practice derivations of motional EMF, self-inductance of solenoid, and impedance in LCR.
• Solve numericals on resonance, power factor, transformer efficiency, and growth/decay in LR circuits.
• Understand phasor diagrams and phase relationships clearly.
• Link with previous chapters: Magnetic field due to current (Chapter 4), motional EMF from Lorentz force.