Electromagnetic Induction— Notes, Formulas & Revision

Magnetic flux Φ_B = ∫B·dA — the 'amount' of B threading a surface.

For a uniform B and a flat loop of area A: Φ_B = B·A·cos θ, where θ is the angle between B and the area-normal n̂.

Unit: weber (Wb) = T·m² = V·s. One weber is a flux that, if collapsed in 1 s, induces 1 V in a single loop.

Φ is maximum when B is parallel to n̂ (loop face perpendicular to B); zero when B is in the plane of the loop.

Flux is a SCALAR, but it has a sign that flips with the chosen normal direction. Conventionally the right-hand rule from the loop's positive sense fixes n̂.

Total flux through a closed surface is ZERO (Gauss's law for magnetism — no magnetic monopoles): ∮B·dA = 0.

Without a changing flux, no EMF is induced — even a coil in a huge constant B has zero EMF.

Faraday's law: the EMF induced in a circuit equals the NEGATIVE rate of change of magnetic flux through it.

For an N-turn coil: ε = −N·dΦ/dt. The minus sign is Lenz's law — encodes opposition.

Any way of changing flux generates EMF: moving the magnet, moving the coil, changing B, deforming the loop, rotating the loop.

Faraday discovered this experimentally (1831) by plunging a magnet through a coil and seeing the galvanometer kick.

EMF is proportional to the RATE of flux change — faster motion ⇒ bigger ε.

Direction of induced current is set by Lenz's law: induced B opposes the change in Φ.

Faraday's law is one of Maxwell's four equations and underlies generators, transformers, induction cooktops, microphones, electric guitars, and metal detectors.

Lenz's law: the direction of induced current is such that it OPPOSES the change producing it.

If flux INTO the loop is increasing, induced current circulates so that its field points OUT of the loop (to oppose the increase).

If flux is decreasing, induced current creates a field in the SAME direction as the (decreasing) original field — trying to maintain it.

This is a direct consequence of CONSERVATION OF ENERGY — if induced current AIDED the change, energy would appear from nothing.

When you push a magnet toward a coil, the induced current makes the coil-face FACING the magnet a repelling pole (same pole). You feel resistance — that mechanical work becomes electrical energy.

Pulling the magnet away makes the facing coil-face attracting (opposite pole) — again the force opposes your motion.

Lenz's law sets the SIGN of the EMF in Faraday's law: ε = −N·dΦ/dt.

When a conductor of length L moves with velocity v through a magnetic field B, the free charges experience a magnetic force F = qv × B that pushes them along the rod.

If v, B, and the rod are mutually perpendicular, charges accumulate at the ends until the resulting E-field cancels the magnetic force.

The end-to-end voltage difference (motional EMF) is ε = BLv.

If the rod is part of a circuit with resistance R: induced current I = ε/R = BLv/R.

The current carrier in the rod (length L, current I) experiences a force F_mag = BIL in the direction OPPOSING its motion — Lenz again.

Power dissipated in the resistor = εI = (BLv)²/R = F_mech · v. All the mechanical work goes to electrical energy.

This is the basic principle behind rail-guns, dynamos, and a generator's armature.

A changing magnetic field produces a non-conservative electric field, even where no charges or wires exist: ∮E·dl = −dΦ_B/dt.

For a cylindrically symmetric region of uniform B(t) of radius R: E inside follows E(r) = (r/2)·|dB/dt| (rises linearly with r).

Outside the region (r ≥ R): E(r) = (R²/(2r))·|dB/dt| — falls off as 1/r.

E is maximum at the boundary r = R and the field lines form closed loops AROUND the axis (just like B around a current — but without any current).

Unlike electrostatic E from charges, this induced E is NOT conservative: ∮E·dl ≠ 0 — there's no scalar potential V for it.

Direction: use the right-hand rule with B(t). If B is INTO the page and INCREASING, induced E circulates COUNTER-CLOCKWISE (so a positive charge would be pushed in the direction that opposes the flux increase).

Induced E is the mechanism behind betatrons (electron accelerators using changing B in a toroidal volume).

When current in a coil changes, its own flux changes, inducing a back-EMF that opposes the change. The proportionality constant is the self-inductance L: ε = −L·dI/dt.

Unit: henry (H) = V·s/A. A 1 H inductor produces 1 V when its current changes at 1 A/s.

For a long solenoid of length ℓ, area A, N turns: L = μ₀·N²·A/ℓ. Adding a ferromagnetic core multiplies by μ_r.

L depends ONLY on geometry and core material — not on the current.

Energy stored in an inductor: U = ½LI². This is energy stored in the MAGNETIC FIELD.

Energy density of the field: u = B²/(2μ₀) — analogous to ε₀E²/2 for an E-field.

In a circuit, inductors resist sudden current changes (just like capacitors resist sudden voltage changes).

Two coils nearby: a changing current in one induces an EMF in the other through shared flux. The constant is the mutual inductance M.

Defining relation: ε₂ = −M · dI₁/dt (and symmetrically ε₁ = −M · dI₂/dt).

Unit: henry (H), same as self-inductance.

Two coaxial solenoids (outer = primary, inner = secondary, same area A and length ℓ): M = μ₀·N₁·N₂·A/ℓ.

M depends only on geometry and core material — never on currents.

M is SYMMETRIC: M₁₂ = M₂₁. EMF induced in coil 2 per dI/dt in coil 1 equals EMF induced in coil 1 per dI/dt in coil 2.

Mutual inductance is the basic principle of transformers, induction cookers, and wireless charging.

Coupling coefficient k = M/√(L₁L₂), with 0 ≤ k ≤ 1. Perfect coupling (k=1) means ALL flux links both coils.

Eddy currents are circulating currents induced in a BULK conductor whenever flux through it changes — not in a wire loop, but in the material itself.

By Lenz's law, they always flow so as to oppose the change in flux — producing a force or torque that resists the motion or change creating them.

They dissipate energy as heat (I²R), which can be a loss or a feature depending on context.

Bad eddy currents: in transformer cores, motor armatures, induction-coil bodies — energy is wasted as heat. Mitigation: LAMINATE the core into thin insulated sheets so eddy paths are short.

Good eddy currents: induction cooktops (heat pots directly), electromagnetic brakes (no contact, no wear, smooth braking), metal detectors (sensing induced eddies), induction furnaces.

A swinging metallic plate between magnet poles is heavily damped by eddy currents — its kinetic energy is converted into heat. Slot the plate and the damping disappears.

Power dissipated by eddy currents in laminated cores scales as (B·f·t)² where t = lamination thickness, so thinner laminations dramatically reduce losses.