Electric Potential and Capacitance

Electric Potential

Potential is the height map of the electric world: charges roll downhill through voltage the way balls roll downhill through altitude. It is a scalar, much easier to compute than the vector field $\vec{E}$.

The electric potential $V$ is the potential energy per unit charge, defined relative to a reference point:

$$V = -\int_{\text{ref}}^{\vec{r}} \vec{E}\cdot d\vec{l}, \qquad \vec{E} = -\nabla V$$

For a point charge, the potential of a point charge is given by:

$$V = k_e\frac{Q}{r}$$

Capacitance & Pitfalls

Capacitance measures how a device stores charge and energy at a given voltage:

$$C = \frac{Q}{V}, \qquad U = \frac{1}{2}CV^2 = \frac{Q^2}{2C}$$

For a parallel-plate capacitor, geometry dictates capacity:

$$C = \varepsilon_0\frac{A}{d}$$

Combine capacitors using these rules:

Common pitfall: Zero potential does not mean zero field, and zero field does not mean zero potential. The field is the slope of $V$; midway between two equal positive charges the field vanishes while the potential remains large.