11.8 Resistor Capacitor (RC) Circuits

An RC circuit contains a resistor and capacitor and exhibits time-dependent behavior. When a capacitor charges through a resistor from a battery ${\mathcal{E}}$, the current is initially largest and then decays to zero as the capacitor fills. During charging, the loop equation is

Since ${I = dq/dt}$, this becomes the differential equation

whose solution is

and therefore

The combination ${\tau = RC}$ is the time constant.

For discharging, the battery is removed and the loop equation becomes

or

with solution

and

The minus sign indicates discharge direction opposite the charging direction convention. These are among the most calculus-heavy equations in the circuits unit because they come from solving first-order differential equations.