Electric Potential and Capacitance

Potential is the "height map" of the electric world: charges roll downhill through voltage the way balls roll downhill through altitude. One number per point (not three, like the field) — which is why circuit analysis speaks in volts, not field vectors.

The electric potential $V$ is the potential energy per unit charge:

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

Potential of a point charge:

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

Key relationships

Capacitance — A capacitor stores charge and energy:

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

Parallel-plate capacitor:

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

Combinations

• Parallel: $C_{\text{eq}} = C_1 + C_2$
• Series: $\dfrac{1}{C_{\text{eq}}} = \dfrac{1}{C_1} + \dfrac{1}{C_2}$

Physics link: Potential is a scalar — much easier to compute than the vector field $\vec{E}$. Find $V$ first, then take the gradient to get $\vec{E}$.

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