Equipotential Lines

Since the electric potential created by a single charge $V=k{q_s\over d}$ depends only on the distance from the charge, it follows that every point a distance $d$ from the charge has the same potential. Those points form a circle around the charge (or a sphere in 3D), and so we can draw a circle around the point charge and label it with the potential that all of those points share. This is called an equipotential line. (In 3D this would be a surface rather than a line.) By looking at multiple equipotential lines, we can gain an understanding of the way space is distorted around the positive charge, and how target charges will react to it.

To do that, we can rearrange the formula from the previous page relating potential energy and potential, to give us the equation

That is, if I place a charge $q_T$ on an equipotential $V$, the charge will have potential energy $q_TV$. For example, placing a $q_T=+1\mu C$ positive charge on the 8V line in the above figure will give that charge an energy of 8µJ, while placing it on the 6V line gives it 6µJ of energy. What would happen if I placed the charge on the 8V line and let go? Remember that systems will always try to reduce their potential energy if possible. Our positive target charge can do this if it moves from the 8V line to the 6V line, and so it does, moving away from the positive source charge: and this is exactly what we expect, that a positive charge will be repelled or pushed away. What if the test charge is negative: $q_T=-1\mu C$? Then it has energy -8µJ on the 8V line and -6µJ on the 6V line. But -6µJ; is higher than -8µJ, and so a negative charge placed on the 8V line won't move to the 6V line. Instead it will move inward to the 12V line (where it has -12µJ of energy), attracted as it is to the positive charge. In short, positive charges move toward lower potential, while negative charges move toward higher potential (given the choice); but both do so in order to decrease their potential energy.

One nice thing about equipotentials is that they allow you to characterize the electric field in a space without knowing or caring about the sources that are creating it. For example, this picture shows three vertical equipotential lines, created by some charges to the left or right of it, positive or negative: we don't know. And yet I can still say that a positive charge placed on the middle line will move to the left (to lower potential), while a negative charge will move to the right.