# Capacitors and Inductors in AC Circuits

fact
The current through a capacitor is given by: $$i_c = C\frac{dV}{dt}$$
example

The voltage across a 1F capacitor is the triangular wave shown below. Find the current running through the capacitor.

This sawtooth wave has peaks at 1 and -1 with a period of $$2\pi$$. The current through the capacitor is given by: $$I = C\frac{dV}{dt}$$ The sawtooth has a slope of $$\pm \frac{2}{\pi}$$ so the derivative is a square wave with peaks at $$\pm \frac{2}{\pi}$$ and frequency $$2\pi$$.
fact
When we apply a sine wave across a capacitor we get a scaled version of that same sine wave but shifted left by $$90^\circ$$ (or "leading" by $$90^\circ$$).
fact
A capacitive circuit has the current leading the voltage.
fact
The voltage across an inductor is given by: $$V_L = L\frac{dI_L}{dt}$$ Which means the current is given by: $$I_L = \frac{1}{L}\int V_Ldt$$
example

The voltage across a 1H inductor is the pulse waveform shown below. Find the current through it.

The voltage pulse has peaks at $$\pm 1$$ and a period of $$2\pi$$. The current is given by our formula: $$I_L = \frac{1}{L}\int V_Ldt$$ So our current is the triangular waveform with peaks at $$\pm \pi$$ and a period of $$2\pi$$.
fact
When we apply a sine wave across an inductor we get a scaled version of that same sine wave but shifted right by $$90^\circ$$ (or "lagging" by $$90^\circ$$).
fact
An inductive circuit has the current lagging the voltage.
practice problems