What Is Electric Potential and How Is It Defined? (HL)

Electric potential at a point is the work done per unit positive test charge in bringing the charge from infinity to that point. It is a scalar quantity, measured in volts, where $1\ \text{V}=1\ \text{J C}^{-1}$.

Definition and mechanism

For a small positive test charge $q$, electric potential is defined by

$$V=\frac{W}{q}$$

where $W$ is the work required to bring the charge from infinity. The reference value is $V=0$ at infinity. Equivalently, electric potential is electric potential energy per unit charge:

$$V=\frac{E_{\text{p}}}{q}$$

For a point charge $Q$, the potential at distance $r$ is

$$V=\frac{kQ}{r}$$

Therefore, potential is positive around a positive source charge and negative around a negative source charge. Because potential is scalar, the total potential produced by several charges is found by algebraic addition:

$$V_{\text{total}}=\sum_i \frac{kQ_i}{r_i}$$

This use of superposition is simpler than adding electric fields: fields are vectors, whereas potentials have no direction.

For example, the potential $0.20\ \text{m}$ from a charge of $+3.0\ \text{nC}$ is

$$V=\frac{(8.99\times10^9)(3.0\times10^{-9})}{0.20}=135\ \text{V}$$

If a $+2.0\ \mu\text{C}$ charge is placed there, its potential energy is $E_{\text{p}}=qV=2.7\times10^{-4}\ \text{J}$. A negative test charge would instead have negative potential energy.

The potential difference between two points is the work done per unit charge in moving a positive test charge between them. Electric field strength is related to the potential gradient:

$$E=-\frac{\Delta V}{\Delta r}$$

The minus sign shows that the electric field points toward decreasing potential. In a uniform field, $E=|\Delta V|/d$. Equipotential surfaces are perpendicular to field lines, and no work is done moving along one.

A common misconception is that zero electric field means zero potential. These conditions are independent: between two equal positive charges, the field is zero at the midpoint, but the potential is positive and non-zero.

Exam technique

This is Additional Higher Level (AHL) content in D.2 Electric and magnetic fields. In calculations, state the zero-at-infinity reference, include charge signs when adding potentials, and remember that potential is scalar rather than vector.