Electromagnetic Oscillations and Alternating Current

We have met three passive circuit elements separately. A resistor dissipates energy; a capacitor stores it in an electric field; an inductor stores it in a magnetic field. In Chapter 30 we watched current rise and fall in an RL circuit, and earlier we watched charge build and drain in an RC circuit. In both, the approach to the final state was a one-way exponential — energy flowed in or out and the story ended.

Put a capacitor and an inductor together with little or no resistance and something far richer happens. Energy no longer settles; it oscillates, pouring out of the capacitor's electric field into the inductor's magnetic field and back again, over and over. This is the seed of every radio tuner, every oscillator, and ultimately of the electromagnetic waves of the next chapter. And once we drive such a circuit with an external sinusoidal source, we arrive at alternating current — the form in which electrical energy is generated, transmitted, and delivered to nearly every wall socket on Earth.

Charge a capacitor and then connect it across an inductor, with no resistance and no battery. At the first instant all the energy sits in the capacitor's electric field, \(U_E = Q^{2}/2C\), and the current is zero. The capacitor begins to discharge through the inductor, but the inductor opposes any sudden change in current, so the current builds gradually. As charge drains, \(U_E\) falls while the magnetic energy \(U_B = \tfrac{1}{2}Li^{2}\) rises.

When the capacitor is fully discharged, the electric field is gone and all the energy has moved into the magnetic field; the current is now maximum. But a current through an inductor cannot stop on a dime — it keeps flowing, now recharging the capacitor with the opposite polarity. The energy slides back into the electric field, the current falls to zero, and the capacitor finds itself charged backwards, ready to discharge the other way. The cycle repeats indefinitely. This back-and-forth is the LC oscillation, and both the charge on the capacitor and the current in the loop vary sinusoidally in time.

The LC oscillation is not merely like a block on a spring — mathematically it is the very same problem. For the spring, energy alternates between kinetic \(\tfrac{1}{2}mv^{2}\) and potential \(\tfrac{1}{2}kx^{2}\). For the circuit, it alternates between magnetic \(\tfrac{1}{2}Li^{2}\) and electric \(q^{2}/2C\). Lining up the two energies term by term reveals the dictionary that translates one system into the other.

The block oscillates at \(\omega = \sqrt{k/m}\). Make the substitutions \(k \to 1/C\) and \(m \to L\) and you instantly read off the angular frequency of the LC circuit — no new derivation needed.

Let us make it rigorous. With no resistor or source, the total energy \(U = U_E + U_B\) is constant, so \(dU/dt = 0\). Writing \(U = q^{2}/2C + \tfrac{1}{2}Li^{2}\) and using \(i = dq/dt\) gives the differential equation of motion.

Differentiating the charge gives the current, which leads the charge by a quarter cycle. The two amplitudes are linked through \(\omega\).

Real circuits always have some resistance. Insert \(R\) in series with \(L\) and \(C\) and the loop dissipates energy at the rate \(i^{2}R\). The total electromagnetic energy now decreases with time, and the oscillation gradually dies away — the exact electrical analogue of a pendulum slowed by air drag. Setting \(dU/dt = -i^{2}R\) yields the damped equation.

To keep an oscillation going forever we must feed energy in to replace what resistance removes. Connect an external alternating emf \(\varepsilon = \varepsilon_m\sin\omega_d t\) — the kind produced by a rotating coil in a generator. After any initial transient dies out, the circuit settles into a steady forced (driven) oscillation: the current alternates at the driving angular frequency \(\omega_d\), not the circuit's natural frequency \(\omega\).

Drive each element alone with the source \(\varepsilon_m\sin\omega_d t\) and watch the phase relationship between the voltage across it and the current through it. A handy mnemonic is "ELI the ICE man": in an inductor (L) the emf E leads the current I, while in a capacitor (C) the current I leads the emf E.

Now put all three in series across the source. Because they carry the same current, we add their voltages as phasors — rotating arrows whose lengths are the voltage amplitudes and whose angles encode the phase. The resistor voltage is along the current; the inductor voltage runs \(90^\circ\) ahead; the capacitor voltage \(90^\circ\) behind. Adding them like vectors and matching the source amplitude gives the current amplitude and phase in one stroke.

Energy is dissipated only in the resistor, and the instantaneous rate \(i^{2}R\) rises and falls through each cycle. What matters in practice is the average rate. Because the average of \(\sin^{2}\) over a cycle is \(\tfrac{1}{2}\), it is natural to define root-mean-square values, which let AC quantities be handled with the familiar DC power formulas.

Power is delivered most efficiently to a resistive load when its resistance is matched to the source, and transmitted over long lines most efficiently at very high voltage and low current (to minimize \(i^{2}R\) losses in the wires). The device that steps voltage up or down at will — using only the mutual induction of Chapter 30 — is the transformer: two coils wound on a common iron core, a primary of \(N_p\) turns and a secondary of \(N_s\) turns.

For the free oscillator, lean on the energy bookkeeping: at any instant \(q^{2}/2C + \tfrac{1}{2}Li^{2}\) equals the constant total, and the peaks satisfy \(\tfrac{1}{2}CV^{2} = \tfrac{1}{2}LI^{2}\). For driven circuits, always compute \(X_L\) and \(X_C\) first, then \(Z\), then \(I\) and \(\phi\) — and remember that power lives only in the resistor.