Electric potential (energy)

Electric potential energy is similar to gravitational potential energy, except it can be positive or negative since the electric force can repel as well as attract. Electric potential energy is given by the formula

Eel=kq1q2r\displaystyle E_{\text{el}} = k \frac{q_{1} q_{2}}{r}.

From this we can derive a formula for a change in electric potential energy:

ΔEel=kq1q2(1r21r1)\displaystyle \Delta{} E_{\text{el}} = k q_{1} q_{2} \left ( \frac{1}{r_{2}} - \frac{1}{r_{1}} \right ).

Electric pontential is, confusingly, not the same as electric potential energy. While the latter is measured in joules, electric potential is measured in joules per coulomb. It represents the Eel\displaystyle E_{\text{el}} that a unitary point charge would have at a particular point. It is denoted with V\displaystyle V and calculated with

V=Eelq2=kq1r\displaystyle V = \frac{E_{\text{el}}}{q_{2}} = \frac{k q_{1}}{r}.

V\displaystyle V is to Eel\displaystyle E_{\text{el}} as ϵ\displaystyle \vec{\epsilon} is to Fel\displaystyle F_{\text{el}}. They differ only by an r\displaystyle r in the denominator:

ϵ=Vr\displaystyle \left \lvert \vec{\epsilon} \right \rvert = \frac{V}{r}.

Most of the time, we talk not about electric potential but about electric potential difference, also known as voltage:

ΔV=kq(1r21r1)\displaystyle \Delta{} V = k q \left ( \frac{1}{r_{2}} - \frac{1}{r_{1}} \right ).

In general, we can relate work to voltage with

W=ΔEel=q2ΔV\displaystyle W = \Delta{} E_{\text{el}} = q_{2} \Delta{} V.

For parallel plates, we have a more specific equation. The work done by the electric force to move a charge q2\displaystyle q_{2} from one plate to the other is given by

W=ΔEel=q2ϵd\displaystyle W = \Delta{} E_{\text{el}} = q_{2} \left \lvert \vec{\epsilon} \right \rvert d,

where d\displaystyle d is the distance between the plates in metres (m). Why were we able to replace ΔV\displaystyle \Delta{} V by ϵd\displaystyle \left \lvert \vec{\epsilon} \right \rvert d? Because, for parallel plates,

ϵ=ΔVd\displaystyle \left \lvert \vec{\epsilon} \right \rvert = \frac{\Delta{} V}{d}.

This is not true for point charges. It looks similar to the general equation ϵ=V/r\displaystyle \left \lvert \vec{\epsilon} \right \rvert = V / r mentioned above, but that Δ\displaystyle \Delta{} makes a big difference. With point charges, the amount of work done is strictly related to the values of r1\displaystyle r_{1} and r2\displaystyle r_{2}. Not so with parallel plates—since the electric field is uniform, it doesn’t matter where the particle is between the plates.