Faraday Law of Induction

Faraday's law of electromagnetic induction is the cornerstone of all electrical energy generation and transformation. It establishes the quantitative relationship between a changing magnetic flux and the electromotive force (emf) induced in a circuit. Without this law, generators, transformers, and induction motors could not be understood or designed.

Core Concept Explanation

Faraday's law states that the electromotive force (emf) induced in any closed loop is equal to the negative rate of change of the magnetic flux through that loop. In equation form: emf = -dΦ/dt. Here Φ = ∫∫ B · dS is the total magnetic flux through the surface bounded by the loop. The negative sign is the mathematical expression of Lenz law: the induced emf always acts to oppose the change that caused it.

The law applies regardless of what causes the flux to change. The flux Φ = B·A·cos(θ) can change because B is time-varying (transformer action), because the area A of the loop changes (generator action), because the orientation angle θ changes (rotating machine), or because of any combination of these. Faraday's law captures all three cases in one equation.

For a coil of N turns, each turn links the same flux Φ, so the total flux linkage is λ = N·Φ and the induced emf is emf = -dλ/dt = -N·dΦ/dt. This is the generalized form used in transformer and inductor analysis. For a coil with inductance L, λ = L·I, and then emf = -L·(dI/dt), recovering the inductor voltage law.

Mathematical Expression

The integral form of Faraday's law relates the line integral of the electric field around a closed path to the surface integral of the time derivative of B. This is the most general form and is one of Maxwell's four equations in integral form.

The closed loop emf is defined as the work done per unit charge around the loop: emf = ∮ E · dl. Faraday's law then states that this equals -d/dt ∫∫ B · dS. The differential (point) form of this law is: curl E = -∂B/∂t. This tells us that a time-varying magnetic field at a point in space produces a rotational electric field at that point. This is fundamentally different from the conservative electric field of static charges.

For the simple case of a planar loop of area A in a uniform time-varying field B(t) = B₀·cos(ωt) perpendicular to the loop: Φ = B·A = B₀·A·cos(ωt). Then emf = -dΦ/dt = B₀·A·ω·sin(ωt). The peak emf is E_peak = N·B₀·A·ω = N·B₀·A·(2πf). This formula is used directly to design the number of turns in AC generators.

Practical Understanding

In a transformer, the primary winding carries an AC current which creates a time-varying flux in the core. This varying flux links the secondary winding, inducing an emf by Faraday's law. The turns ratio determines the voltage transformation: V2/V1 = N2/N1. The transformer works purely on Faraday's law with no moving parts.

In an AC generator, a coil rotates in a static magnetic field. The angle between the coil normal and B changes sinusoidally, changing the flux linkage and inducing a sinusoidal emf. The peak emf depends on the rotation speed ω, the number of turns N, the area A, and the field strength B. This is the operating principle of all power station generators.

Given:
Coil: N = 100 turns, area A = 50 cm² = 50×10⁻⁴ m²
Magnetic field: B = 0.2·sin(100πt) T (time-varying, perpendicular to coil)

Why this formula applies:
B is time-varying and coil is stationary. Use emf = -N·dΦ/dt.

Formula:
Φ = B · A = 0.2·sin(100πt) × 50×10⁻⁴
Φ = 1×10⁻³·sin(100πt) Wb
emf = -N · dΦ/dt

Substitution:
dΦ/dt = 1×10⁻³ × 100π × cos(100πt)
dΦ/dt = 0.1π·cos(100πt)

Calculation:
emf = -100 × 0.1π·cos(100πt)
emf = -10π·cos(100πt) V

Peak emf = 10π ≈ 31.4 V

Final Answer: emf = -31.4·cos(100πt) V. Peak induced emf is 31.4 V at t = 0.

Exam Tip: In GATE problems, carefully check whether B is given as a function of time or space. If B = f(t) at a stationary loop, use emf = -N·A·dB/dt. If B is uniform and the loop moves, use emf = B·l·v for a straight conductor moving with velocity v in field B. These are two different scenarios but both follow from emf = -dΦ/dt.

Mechanism of Induced emf

• Transformer emf: stationary coil, time-varying B. emf = -N·A·(dB/dt) for uniform B over area A.
• Motional emf: moving conductor in static B. emf = B·l·v for conductor of length l moving at velocity v perpendicular to B.
• Rotating machine: both B and geometry combine. Peak emf = N·B·A·ω for coil rotating at angular speed ω.
• The differential form curl E = -∂B/∂t shows that a changing B creates a solenoidal (rotational) E field at every point in space.
• For a static B field, curl E = 0 and E is conservative (no induced emf). The distinction is critical in electromagnetic field theory.

Quick Revision

• Faraday's law: emf = -dΦ/dt. For N turns: emf = -N·dΦ/dt = -dλ/dt.
• Flux: Φ = ∫∫ B · dS. For uniform B over area A at angle θ: Φ = B·A·cos(θ).
• Motional emf: emf = ∮(v × B) · dl = B·l·v (for straight conductor, v ⊥ B ⊥ l).
• Differential form (Maxwell): curl E = -∂B/∂t. This is the third Maxwell equation.
• Trap: The negative sign is part of the law. Forgetting it means you get the wrong direction of induced current.
• Generator peak emf: E_peak = N·B·A·ω. Know this for rotating machine problems.
• Transformer principle: V2/V1 = N2/N1 follows directly from Faraday's law applied to each winding.