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Capacitors

  As you know, a capacitor stores energy by separating a charge $\pm q$, say on two parallel plates. The voltage v across the capacitor is related to the charge by

q = C v

where C is the capacitance. The sign convention is given in Figure 20.

  

Figure 20: Capacitor.

Positive current flows into the + terminal, and is given by

$$i = C \frac{dv}{dt} .$$ (17)

Note that the current flowing into the capacitor equals the current flowing out, and the net charge stored is zero.

Equation (17) is Ohm's law for capacitors.  

If v is constant, then i=0, so a capacitor acts like an open circuit to DC.

In order for current to flow, the voltage must be changing.

Integrating, we have

$$\begin{array}{rl} v(t) & = \frac{1}{C} \int_0^t i(\tau)d\tau ... ...\ \\ & = \frac{1}{C} \int_{-\infty}^t i(\tau)d\tau \end{array}$$

with the convention $v(-\infty)=0$.

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