PHYS 2112 Ch 14 – Inductance, LC and RLC Circuits

Mutual inductance M

M = N₂Φ₂₁/I₁ = N₁Φ₁₂/I₂ — coupling between two nearby circuits

Mutually induced emf

ℰ₂ = −M dI₁/dt ; ℰ₁ = −M dI₂/dt

Self-inductance L

Emf induced in a circuit by its own changing current: ℰ = −L dI/dt ; L = NΦ_B / I

Unit of inductance

Henry (H) = 1 V·s/A

Inductor

Circuit element that provides self-inductance; opposes CHANGES in current

Self-inductance of a solenoid

L = μ₀N²A / l

Energy stored in an inductor

U = ½LI²

Energy density of a magnetic field

u_B = B² / (2μ₀)

RL circuit time constant

τ_L = L/R

RL current growth (switch closed)

I(t) = (ℰ/R)(1 − e^(−t/τ_L))

RL current decay

I(t) = I₀ e^(−t/τ_L)

Voltage across inductor while current grows

V_L = ℰ e^(−t/τ_L) (equals ℰ at t = 0, → 0 as t → ∞)

Inductor at t = 0 vs t → ∞ (DC)

t = 0: acts like an open circuit (I = 0); t → ∞: acts like a plain wire

Large vs small τ_L

Large τ_L → current changes slowly; small τ_L → current changes quickly

LC circuit equation

Q/C + L dI/dt = 0

LC charge solution

Q(t) = Q_max cos(ωt + φ)

LC angular frequency

ω = 1/√(LC)

Phase between charge and current in LC

90° out of phase — when Q is max, I = 0 and vice versa

LC energy

Energy sloshes between capacitor (Q²/2C, electric) and inductor (½LI², magnetic); total is conserved

Voltage across inductor in LC at t = 0 (capacitor full)

V_L = Q_max / C

Inductor as a mechanical analogy

Like mass (inertia)/spring — stores energy and gives it back without loss

RLC circuit

Like LC, but resistor dissipates energy → damped oscillations

RLC charge (underdamped)

Q(t) = Q_max e^(−γt) cos(ω′t + φ)

Damping factor γ

γ = R / (2L); how fast oscillations die out

Damped oscillation frequency

ω′² = ω₀² − γ², with ω₀ = 1/√(LC)

Bandwidth (RLC)

Δω = R/L — range of frequencies near resonance where the circuit responds strongly

Quality factor Q

Q = (1/R)√(L/C) = ω₀/Δω — sharpness of the resonance

Applications of LC / RL / RLC circuits

LC: radio tuning, signal generators, filters; RL: metal detectors; RLC: MRI