17.4 A Simple Electric Circuit

If we use wires to connect simple devices together we can form an electric circuit. Such circuits can be used for all sorts of purposes. Their study took off in the development of radios, a hundred years ago.

One simple circuit consists of four elements as follows: there is a power source which produces a difference of potential between its two terminals; a coil, which consists of a winding of wire, a gap in the wire and perhaps a device that offers resistance to the flow of electrons in it. Electrons flow through the circuit and pile up on the side of the gap. If we represent the total charge that does so as \(q(t)\), in some unit, the current, \(i(t)\) that flows in the circuit is \(q'(t)\). The resistance \(R\) of the circuit by Ohm’s law, produces a difference of potential of \(Ri(t)\), the difference of potential across the gap is \(\frac{q(t)}{C}\) where \(C\) is called the capacity of the gap. Changes of current, according to Faraday’s law, cause a difference of potential across the coil of \(Li'(t)\), for some constant \(L\) and so if the power source produces a difference of potential \(Vsin(\omega t)\) we find that this system obeys the equation

\[ \begin{aligned} Vsin(\omega t) &= Ri(t) + \frac{q(t)}{C} + I'(t)L \\ \quad &= Rq'(t) + \frac{q(t)}{C} + q''(t)L \end{aligned} \]

This is exactly the same as the equation of the forced harmonic oscillator, and has all the same consequences.