Electromagnetic Induction

Magnetic flux through a surface S is the integral of the normal component of B over the surface. For a uniform field and flat surface: Φ = BA cosθ, where θ is the angle between B and the area normal. SI unit: weber (Wb = T·m²).

Total flux linkage for an N-turn coil is NΦB. For a linear inductor, NΦ = LI where L is self-inductance. SI unit: Wb (same as volt·second). Flux linkage is the quantity whose rate of change determines induced EMF.

Induced EMF equals the negative rate of change of flux linkage. The negative sign embeds Lenz's law. The larger the rate of flux change and the more turns, the greater the induced EMF. Applies regardless of the mechanism causing flux change.

The induced current always flows in a direction to oppose the change in flux that caused it. This is a statement of energy conservation — you must do work against the opposing force to sustain the change. The negative sign in Faraday's law is its mathematical expression.

EMF induced in a straight conductor of length l moving with velocity v in field B. θ is the angle between v and B (or equivalently between l and (v×B)). Maximum when v ⊥ B and l ⊥ both. Origin: Lorentz force on free charges in the rod.

General expression for motional EMF in any moving conductor. The integrand (v×B) is the non-electrostatic force per unit charge (motional electric field). Used for rotating rods, irregularly shaped conductors, or non-uniform fields.

EMF induced in a rod of length l rotating with angular velocity ω about one end in a uniform field B perpendicular to the plane of rotation. Here v = ωl is the tip velocity. Derived by integrating the motional EMF over each elemental length dr: dε = Bωr dr.

Total charge induced is independent of how fast the flux changes — depends only on the total flux change and resistance. q = ∫i dt = ∫(ε/R) dt = ΔΦ/R. This principle is used in fluxmeter/ballistic galvanometer to measure flux.

Self-inductance L is the flux linkage per unit current. Induced back-EMF opposes the change in current (Lenz's law). SI unit: henry (H = Wb A⁻¹ = V·s·A⁻¹). L depends only on geometry; it is a property of the coil, not the current.

For a solenoid of N turns, length l, cross-section A, and turn density n = N/l. With a core of relative permeability μr: L = μ₀μr n²Al. L scales as N² — doubling turns quadruples inductance. Assumes uniform field inside (l ≫ √A).

Inductance of a toroid with N turns, mean radius r, and cross-sectional area A. Field is confined entirely inside the toroid (zero outside). With core: L = μ₀μrN²A/(2πr). The toroid is a solenoid bent into a closed ring.

Series: +2M for aiding (fluxes add), −2M for opposing (fluxes subtract). Parallel formula assumes no mutual induction. These are analogous to resistance combinations but with the added complication of mutual inductance when coils are nearby.

Mutual inductance M is the flux linkage in coil 2 per unit current in coil 1. By Neumann's formula, M₁₂ = M₂₁ (reciprocity). EMF induced in coil 2 when current in coil 1 changes. SI unit: henry (H). Basis of transformer operation.

For two coaxial solenoids: outer solenoid (n₁ turns/m, length l) and inner coil (N₂ turns, area A). M depends on geometry only, not on which coil drives current. With a magnetic core: M = μ₀μrn₁N₂A.

k measures the fraction of flux from coil 1 that links with coil 2. k = 1 (perfect coupling, all flux shared), k = 0 (no coupling). Ideal transformer assumes k = 1. Real transformers: k ≈ 0.95–0.99. Also: M = k√(L₁L₂).

Energy stored in the magnetic field of an inductor carrying current I. Analogous to ½CV² for capacitor. This energy is stored in the magnetic field and is recovered when the current decreases. SI unit: joule (J).

Energy per unit volume stored in a magnetic field B in free space. In a medium with permeability μ: u = B²/(2μ). Analogous to electric field energy density u_E = ½ε₀E². Used to compute total energy in solenoids and other field regions.

Current builds exponentially toward steady state I₀ = ε/R when a battery is connected. τ = L/R is the inductive time constant. At t = τ: I = 0.632 I₀. At t = 5τ: I ≈ 0.993 I₀ (effectively steady). Back-EMF = L dI/dt decreases as current rises.

When battery is removed, current decays exponentially. Energy stored in inductor (½LI₀²) is dissipated in R. At t = τ: I = 0.368 I₀. Sudden breaking of circuit causes large dI/dt, hence large induced EMF — this is why inductors cause sparking at switch-off.

Natural frequency of oscillation in an ideal LC circuit (no resistance). Charge and current oscillate sinusoidally. Energy alternates between electric (capacitor) and magnetic (inductor) forms. Analogous to simple harmonic motion: L → mass, 1/C → spring constant.

Charge oscillates as cosine; current (I = −dq/dt) as sine — they are 90° out of phase. Maximum current I₀ = q₀ω₀ = q₀/√(LC). Current is maximum when charge is zero (all energy in inductor) and vice versa.

Total electromagnetic energy is conserved in an ideal (lossless) LC circuit. Electric energy q²/2C and magnetic energy ½LI² oscillate in antiphase but their sum remains constant. In a real circuit with resistance R, energy decays as e^(−Rt/L).