Electric Potential

Source lecture(s): SC134 Lec5

Intuition

Potential is the potential-energy cost per unit charge—a scalar that is often easier to compute than the vector field.

Formal Definition

\[V = \frac{U}{q}$$ From field: $$\Delta V = -\int \vec{E}\cdot d\vec{s}\]

Mathematical Formulation

Point charge: $V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$ Group: $V = \sum \frac{1}{4\pi\varepsilon_0}\frac{q_i}{r_i}$ Continuous: $V = \int \frac{1}{4\pi\varepsilon_0}\frac{dq}{r}$

Derivation

Work to bring $q_0$ from infinity to $P$ against conservative force; divide by $q_0$.

Worked Example

Dipole on axis at distance $r\gg d$: $V\approx(1/4\pi\varepsilon_0)p/r^2$.

Common Mistakes

Sign errors when going from $E$ to $V$ (remember $-$).
Forgetting reference at infinity.

Quiz

Q1. What are the units of potential?

Answer

Volts (J/C).

Q2. Is potential a scalar or vector?