Inductance and LC Oscillations
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Direct answer
Energy U = (1/2) L I^2 sits stored in the magnetic field of a current-carrying inductor, and L, defined by flux linkage N phi = L I, opposes change: emf = −L dI/dt, so current in an L–R circuit cannot jump, growing as I(0)(1 − e^−t/tau) on switch-on and decaying as I(0) e^−t/tau after, with time constant tau = L/R. Pair an inductor with a capacitor and the charge oscillates at f = 1/(2 pi sqrt(LC)), with energy sloshing between capacitor (q^2/2C) and inductor (L I^2/2) — the exact electrical twin of a spring–mass oscillator, where q plays x, I plays v, 1/C plays k and L plays mass.
What you must remember
- Self and mutual inductance: L = N phi/I with emf = −L dI/dt; mutual M = N(2) phi(2)/I(1) with M = k sqrt(L(1) L(2)), k between 0 and 1.
- Magnetic energy: U = (1/2) L I^2, with energy density B^2/(2 mu(0)) — doubling current quadruples stored energy.
- LR transients: growth I = I(0)(1 − e^−t/(L/R)) and decay I(0) e^−t/(L/R); at t = tau the current has covered 63% of its final value; the inductor behaves as a broken wire at switch-on and as a plain wire at steady state.
- LC oscillations: q = q(0) cos(omega t) with omega = 1/sqrt(LC); the current leads by a quarter cycle, peaking when the capacitor is empty.
- The SHM dictionary: q ↔ x, I = dq/dt ↔ v, 1/C ↔ k, L ↔ m, (1/2)q^2/C ↔ potential energy, (1/2)L I^2 ↔ kinetic energy; frequency = (1/2pi) sqrt(k/m) maps to (1/2pi) sqrt(1/LC).
- LC reality: resistance damps the swing, and the stored energy eventually dissipates; sustained oscillations need periodic re-excitation — the seed idea of the LC oscillator.
- Phase in AC: across a pure inductor, voltage leads current by 90 degrees; inductive reactance X(L) = omega L grows with frequency.
- Pattern note: Main tests energy, time constant and LC frequency substitutions; Advanced tests the analogy dictionary quantitatively and LR circuits with two loops.
Watching energy change its costume
Charge a capacitor to q(0) in an LC loop and close the switch. At t = 0, all energy is electrical: U = q(0)^2/(2C), current zero. A quarter cycle later the capacitor is empty and current peaks at I(max) = q(0) sqrt(1/LC), with all energy now magnetic. Put numbers on it: a 2 microfarad capacitor charged to 100 V holds q(0) = 2 × 10^-4 C and U = 0.01 J; with L = 0.5 henry, omega = 1/sqrt(LC) = 10^3 rad/s, f ≈ 159 Hz, and I(max) = q(0) omega = 0.2 A. The single conservation statement q(0)^2/(2C) = (1/2)L I(max)^2 produces the peak current without solving any differential equation.
The LR transient deserves the same one-line honesty. At the instant of switching, the inductor enforces current continuity — I cannot jump — so the full battery voltage appears across L and zero across R; long after, L is an ordinary wire, and tau = L/R is the time to cover 63% of the journey in either direction.
Where students slip
The inductor's voltage is L dI/dt, not L I — at peak current in an LC swing the inductor voltage vanishes while its energy is maximum; conflating the two readings garbles every phasor question. Second, stored energy is (1/2)L I^2, and questions that double I expect a fourfold answer; the same quadratic hides in B^2/(2 mu(0)) density. Third, in the analogy, 1/C maps to k (not C), so a stiffer spring is a smaller capacitance — candidates who pair C with k produce frequencies inverted in the wrong place. Fourth, at t = 0 after switch-on in an LR circuit, the entire source voltage is across the inductor; writing Ohm's law across R at t = 0 (giving I = V/R) is the classic error, because I(0) = 0 strictly.
Frequently asked questions
What is self-inductance physically?
The flux linkage per unit current, L = N phi/I, which makes the circuit oppose changes in its own current through emf = −L dI/dt; it is the inertia of electric current.
Where does (1/2) L I^2 come from?
Work done by the source against the induced emf while the current grows, stored entirely in the magnetic field at energy density B^2/(2 mu(0)).
What is the time constant of an LR circuit?
tau = L/R: the time in which switch-on current reaches 63% of its final value, or decay current falls to 37%; current can never change instantaneously through an inductor.
At what frequency does an LC circuit oscillate?
f = 1/(2 pi sqrt(LC)), independent of amplitude; the charge oscillates like the displacement of a spring–mass system.
How is the LC circuit analogous to SHM?
Charge plays displacement, current plays velocity, inductance plays mass, and 1/C plays the spring constant, so q = q(0) cos(omega t) with omega = 1/sqrt(LC) mirrors x = A cos(omega t).