Capacitors

Made of two conductive plates separated by a dielectric layer.

In an ideal capacitor, the charge imbalance $Q$ is proportional to the voltage $V$ across the plates.

$$Q = CV$$

The proportionality constant is the capacitance ($C$), in terms of physical dimensions: $$

$$C = \frac{\epsilon A}{d}$$

Here:

• $\epsilon$: permittivity of the material in-between
• $d$: distance between the plates
• $A$: area of a plate

v and i

Current $i$ passing through the capacitor and the voltage $v$ across the capacitor are related by:

$$i = C \frac{\text{d}v}{\text{d}t}$$

Note

When 2 capacitors are connected in series, the charge $Q$ is the same for both capacitors. $$

When 2 capacitors are connected in parallel, the voltage $V$ is the same for both capacitors. $$

Total charge is conserved in both cases.

Energy stored

Suppose voltage across an initially uncharged capacitor rises from $0$ to $V$ during a time period of $t$.

$$e = \int_{0}^{t} p\,dt = \int_{0}^{t} vi\,dt = C \int_{0}^{v} v\,dv$$ $$E = \frac{1}{2}CV^2$$