Electromagnetic Induction (AP Physics C: E&M Unit 5) — Concepts, Signs, and Circuit Applications

Electromagnetic induction

Production of an induced emf (and often an induced current) due to a changing magnetic environment—specifically, a change in how magnetic field “links” through a loop over time.

Induced emf (electromotive force)

A voltage around a closed loop created by changing magnetic flux (or equivalently, an induced electric field); measured in volts (V).

Magnetic flux (Φ_B)

A measure of how much magnetic field passes through a surface; for a uniform field through a flat loop, Φ_B = BA cosθ (units: weber, Wb).

Area vector

A vector perpendicular to a surface with magnitude equal to the surface area; used in flux as the direction that defines the angle θ with the magnetic field.

Flux angle (θ) in Φ_B = BA cosθ

The angle between the magnetic field direction and the loop’s area vector (not the angle between the field and the plane of the loop).

Flux integral definition

For non-uniform fields or curved surfaces, magnetic flux is Φ_B = ∫ B⃗ · dA⃗ , where dA⃗ is an infinitesimal area vector.

Faraday’s Law

The induced emf around a closed loop equals the negative time rate of change of magnetic flux: ℰ = − dΦ_B/dt.

Faraday’s Law (N-turn coil)

For a coil with N tightly wound turns experiencing the same flux, the total induced emf is ℰ = −N dΦ_B/dt.

Maxwell–Faraday equation (integral form)

Relates induced electric field to changing flux: ∮ E⃗ · dl⃗ = − dΦ_B/dt; shows emf can exist due to an induced field even without a wire.

Non-conservative electric field

An induced electric field whose field lines form closed loops (not arising from a static potential difference); associated with changing magnetic flux.

Flux-change mechanisms (BAcosθ)

Flux can change if any of B (field strength), A (loop area), or θ (orientation/rotation) changes with time.

Motional emf

An induced emf produced when a conductor moves through a magnetic field so charges feel magnetic force; a special case consistent with Faraday’s Law.

Magnetic force on a charge in a moving conductor

Force causing charge separation in motional emf: F⃗ = q v⃗ × B⃗.

Motional emf formula (rod)

For a straight rod of length L moving at speed v perpendicular to uniform B, the motional emf magnitude is ℰ = BLv.

Lenz’s Law

Direction rule for induction: the induced current produces a magnetic effect that opposes the change in magnetic flux that caused it (the meaning of the minus sign in Faraday’s Law).

“Opposes the change in flux” (key wording)

Correct Lenz’s Law statement: induced effects oppose the change in flux, not necessarily the original magnetic field itself.

Right-hand rule for loop field/current

Curl fingers in the direction of conventional current around a loop; thumb points in the direction of the loop’s induced magnetic field through the loop.

Magnetic force on a current-carrying wire

Force on a wire segment in a magnetic field: F⃗ = I L⃗ × B⃗ (often produces magnetic drag opposing motion in induction setups).

Magnetic drag (induction braking force)

A resistive force on moving conductors due to induced currents/fields (via Lenz’s Law), requiring external work to maintain constant speed.

AC generator (rotating coil model)

A rotating coil in a uniform magnetic field produces changing flux via changing angle, creating a sinusoidal induced emf.

Peak emf of a rotating coil

For a coil rotating with angular speed ω, the maximum induced emf is ℰ_max = NBAω.

Transformer turns ratio (ideal coupling)

If both coils link the same changing flux, induced emf ratio equals turns ratio: ℰs/ℰp = Ns/Np (often written as Vs/Vp = Ns/Np for ideal AC transformers).

Inductor emf law

Self-induced emf in an inductor opposing changes in current: ℰ_L = −L dI/dt (L in henries, H).

RL circuit time constant (τ)

Characteristic time for current to grow/decay in an RL circuit: τ = L/R.

Eddy currents

Circulating currents induced in a bulk conductor by changing magnetic flux; they oppose the change (Lenz’s Law) and often cause heating and magnetic braking.