What Is Electric Potential Energy for Two Charges? (HL)

Electric potential energy for two charges is the energy stored because of their positions in an electric field. For two point charges, it is

$$E_{\text{p}}=\frac{kq_1q_2}{r}$$

where $k=8.99\times10^9\ \text{N m}^2\text{ C}^{-2}$, $$ and are the charges, and is their separation.

The Reasoning

This is an Additional Higher Level (AHL) concept in D.2 Electric and magnetic fields. Electric potential energy is defined as the work required to assemble the system by bringing the charges from infinite separation to their final separation, without changing their kinetic energy.

The zero of potential energy is therefore defined at infinity. Because potential energy is a scalar quantity, the signs of the charges must be included directly in the calculation.

It can also be calculated from electric potential: $E_{\text{p}}=qV$, where $V$ is the potential produced by the other charge. Substituting $V=kq/r$ gives the two-charge equation above.

For example, let $q_1=+2.0\ \mu\text{C}$, $q_2=-3.0\ \mu\text{C}$, and :

$$E_{\text{p}}=\frac{(8.99\times10^9)(2.0\times10^{-6})(-3.0\times10^{-6})}{0.20}=-0.27\ \text{J}$$

The negative value means energy is required to separate the charges to infinity. The electric field does work according to $W_{\text{field}}=-\Delta E_{\text{p}}$.

If the separation decreases while the charges attract, $E_{\text{p}}$ becomes more negative and the electric field transfers energy into kinetic energy. If like charges are moved closer, $E_{\text{p}}$ increases because an external force must do positive work against repulsion.

A common misconception is that negative potential energy means the system has “negative energy.” It instead means the system has less potential energy than the chosen zero at infinity.

Exam Technique

In a Calculate question, show the equation, convert all charges to coulombs, substitute signed values, and interpret the sign. State that $r$ is the centre-to-centre separation. Do not confuse electric potential energy $E_{\text{p}}$, measured in joules, with electric potential $V$, measured in volts.