Unit 5: Electromagnetism

Electromagnetic induction

The production of an electromotive force (emf) in a conductor due to a changing magnetic flux through a loop (changing flux is the key idea; motion is just one way to change flux).

Electromotive force (emf), ℰ

Voltage around a closed loop (work per charge around the loop), often produced by induction rather than a battery.

Magnetic flux, Φ_B

A measure of how much magnetic field passes through a surface; induction depends on changes in Φ_B.

Flux surface integral

General definition of magnetic flux: Φ_B = ∫(surface) B · dA.

Area vector, A (or dA)

A vector perpendicular (normal) to a surface with magnitude equal to the area element; sets the sign/direction of flux in dot products.

Uniform-field flux formula

For a uniform B over a flat area: Φ_B = BAcosθ, where θ is the angle between B and the surface normal (area vector).

Weber (Wb)

SI unit of magnetic flux; 1 Wb = 1 T·m².

Faraday’s law (emf form)

A changing magnetic flux induces an emf: ℰ = − dΦ_B/dt.

Faraday’s law for N turns

For a coil of N identical turns: ℰ = −N dΦ_B/dt (emf scales with the number of turns).

Faraday’s law (integral form)

Maxwell-Faraday equation: ∮ E · dℓ = − dΦ_B/dt.

Induced electric field

An electric field created by a changing magnetic flux; it can “circulate” around a loop even without a battery.

Nonconservative electric field

An electric field with nonzero closed-loop integral (∮ E·dℓ ≠ 0), typical of induction; not describable purely by a single-valued electric potential.

Lenz’s law

The induced emf/current acts to oppose the change in magnetic flux that produced it (required by energy conservation).

Loop right-hand rule

For a current loop: curl fingers in the current direction; thumb points in the direction of the loop’s magnetic field through the loop.

Induced current

A current driven by an induced emf, occurring only if there is a closed conducting path.

Closed conducting path

A complete loop of conductor; needed for sustained induced current (an emf can exist even if the circuit is open, but no continuous loop current flows).

Motional emf

An emf produced when a conductor moves through a magnetic field and charges experience magnetic forces that separate them, creating a potential difference.

Magnetic force on a charge

Force on a moving charge in a magnetic field: F_B = q v × B.

Motional emf of a sliding rod

For the standard perpendicular geometry: ℰ = Bℓv (B field, rod length ℓ, speed v).

Magnetic force on a current-carrying conductor

Force on a current segment in a magnetic field: F = I ℓ × B (magnitude F = IℓB when perpendicular).

Inductor

A circuit element (often a coil) whose changing current changes magnetic flux linkage, producing an induced emf that opposes current changes.

Self-inductance, L

A property of a circuit/coil quantifying how strongly its own changing current induces an opposing emf.

Flux linkage

Total linked flux for a coil, typically written NΦ_B (sum of flux through all turns).

Inductance definition (flux linkage relation)

Defines self-inductance: NΦ_B = LI (valid in the ideal linear regime where linkage is proportional to current).

Henry (H)

SI unit of inductance; 1 H = 1 Wb/A = 1 V·s/A.

Inductor back emf

Induced emf in an inductor: ℰ_L = −L dI/dt (opposes the change in current).

Long-solenoid field

Approximate uniform field inside a long solenoid: B = μ0 n I, where n is turns per length.

Long-solenoid inductance

Inductance of an ideal long solenoid: L = μ0 N²A/ℓ (N turns, area A, length ℓ).

RL circuit

A circuit containing a resistor and inductor; exhibits exponential current growth/decay due to the inductor opposing changes in current.

RL time constant, τ

Timescale for RL transients: τ = L/R.

RL current growth

After closing a switch on a DC source: I(t) = (ℰ/R)(1 − e^(−t/τ)).

RL steady-state current

Long-time current in a DC series RL circuit (ideal inductor): I_∞ = ℰ/R.

RL current decay

After removing the source but keeping a closed RL loop: I(t) = I0 e^(−t/τ).

Inductive voltage spike

A potentially large voltage that can occur when interrupting current in an inductive circuit, since the inductor resists rapid changes in current.

Magnetic energy stored in an inductor

Energy stored in the inductor’s magnetic field: U_B = ½LI².

Magnetic energy density

Energy per volume in a magnetic field (vacuum): u_B = B²/(2μ0).

Mutual inductance, M

A measure of how strongly changing current in one coil induces an emf in a nearby coil via shared changing magnetic flux.

Mutual induction (emf relation)

Induced emf in coil 2 from changing current in coil 1: ℰ2 = −M dI1/dt (and symmetrically ℰ1 = −M dI2/dt).

Transformer

A device that uses mutual inductance (typically with AC) to transfer electrical energy between primary and secondary coils.

Ideal transformer voltage ratio

For an ideal transformer: Vs/Vp = Ns/Np (secondary/primary voltages equal turns ratio).

Ideal transformer power conservation

In an ideal transformer: VpIp = VsIs (input power equals output power).

Transmission line resistive loss

Power lost as heat in transmission lines: P_loss = I²R; higher voltage transmission reduces I for the same power, reducing losses.

LC circuit

An inductor-capacitor circuit where energy oscillates between the capacitor’s electric field and the inductor’s magnetic field (ideal case: negligible resistance).

LC angular frequency, ω

Natural angular frequency of an ideal LC oscillator: ω = 1/√(LC).

LC period, T

Oscillation period of an ideal LC circuit: T = 2π√(LC).

Capacitor energy (in LC context)

Energy stored in a capacitor: U_C = Q²/(2C) (also equals ½CV²).

Total energy in an ideal LC circuit

Energy is conserved and swaps between forms: UC + UL = constant, where U_L = ½LI².

Gauss’s law (electric)

Electric flux through a closed surface: ∮ E · dA = Q_enc/ε0.

Gauss’s law for magnetism

Magnetic flux through any closed surface is zero: ∮ B · dA = 0 (no magnetic monopoles in this model).

Ampere–Maxwell law

Circulation of B around a loop: ∮ B · dℓ = μ0 Ienc + μ0ε0 dΦE/dt (includes the displacement current term).