Electric potential is the electrical potential energy per unit charge, as you learned on page 1048. But what is the electrical potential energy for a given set of electric charges? The answer can be found by considering how much work is required to put those charges into their positions. This connection between work and energy should remind you of the work–energy theorem!

Work done in an electric field

What is the electric potential energy of an electric charge q_{2} in the vicinity of another charge q_{1}? We will answer this question using work and energy by imagining how much work has to be done to move q_{2} to its location. From equation (18.4), the energy of a charge in an electric potential is E_{p} = qV.

When the two charges are far away from each other, r becomes very large and the potential V goes to zero from equation (18.5). The energy likewise goes to zero when the charges are far apart. When the charges are close, however, the energy is E_{p} = qV = k_{e}q_{1}q_{2}/r.

Electric potential energy

The difference between the final energy and the initial energy is the work done to assemble the system—and is the electrical potential energy of the assembled system. The work done to move the charge into place provides an equation for the electric potential energy of a pair of point charges. It takes work to move two positive charges together, so the electric potential energy for two positive charges is positive in equation (18.6). It takes work to separate a positive and a negative charge, so their electric potential energy is negative. Electric potential energy can be thought of as being stored in their mutual electric field.

(18.6)

$${E}_{p}=\frac{{k}_{e}{q}_{1}{q}_{2}}{r}$$

E_{p}

=

electric potential energy (J)

k_{e}

=

Coulomb constant = 9.0×10^{9} N m^{2}/C^{2}

q_{1}

=

electric charge #1 (C)

q_{2}

=

electric charge #2 (C)

r

=

separation (m)

Electric potential energy for two charges

Parallel and perpendicular to electric field

Moving two charges together or apart is equivalent to saying that they are being moved along (or parallel to) the electric field lines. It takes work to move a charged particle along an electric field line. How about if you moved a charged particle along an equipotential line? Since the electric potential does not change along an equipotential line, the electric potential energy also does not change from equation (18.4). Since equipotential lines are perpendicular to the electric field, no work is done against the electric force to move perpendicular to the electric field.