RC circuit and wave transmission: circuits, waves and energy

Problem

A capacitor of capacitance $C = 10\;\mu$F, initially charged to $V_0 = 100$ V, discharges through a resistance $R = 100\;\Omega$. The circuit also includes an antenna that is supposed to radiate an EM wave at frequency $f = 1$ MHz. Determine: (a) the discharge time constant $\tau$, (b) the energy initially stored, (c) discuss the analogy between the RC discharge and the exponential decay of a damped wave, and why the circuit cannot generate the required wave.

Solution

(a) Time constant. For an RC circuit, the characteristic discharge time is the product of resistance and capacitance: $\tau = RC = 100 \times 10\cdot 10^{-6} = 10^{-3}\;\text{s}$ $\ev{\tau = 1\;\text{ms}}$

(b) Stored energy. The energy in a charged capacitor is $E = \tfrac12 CV^2$: $E_0 = \tfrac12 C V_0^2 = \tfrac12\cdot 10\cdot 10^{-6}\cdot (100)^2$ $\ev{E_0 = 0{,}05\;\text{J}}$

(c) Analogy and the circuit’s limitation. The voltage decays exponentially: $V(t) = V_0\,e^{-t/\tau}$

Exponential discharge of the capacitor: after a time $\tau$ the voltage has dropped to 37% of its initial value.

The typical discharge frequency is $\sim 1/\tau \approx 1$ kHz, a thousand times lower than the megahertz required by the antenna. The RC circuit is too slow to modulate a radio signal: to generate MHz waves you need a resonant LC circuit, in which the energy oscillates back and forth between the capacitor and the inductor instead of dissipating.

Universal analogy. The form $V(t)=V_0 e^{-t/\tau}$ is identical to the envelope of a damped wave and to the law $N(t)=N_0 e^{-\lambda t}$ of a population of radioactive nuclei. Exponential decay is a mathematical structure that recurs whenever a quantity decreases at a rate proportional to its own value.

The connections: RC circuits (ch. 15), EM waves and resonant circuits (ch. 19), with analogies to radioactive decay (ch. 22) and the damped oscillator (ch. 4b).

Links

Topics: Electric circuits · Electromagnetic waves Concepts: RC circuit · Capacitance and the capacitor · Radioactive decay