Equipotential Surfaces

Consider a dipole, with charges \(q\) and \(-q\). If \(V_\infty=0\), where else is the potential zero? According to the formula, $$V=k{q\over d_1}+k{-q\over d_2}+V_\infty=kq\left(\frac{1}{d_1}-\frac{1}{d_2}\right)$$ and this is zero at any point which is equidistant from the two charges (where \(d_1=d_2\)). The spot directly between the two charges satisfies that condition, so the potential is zero there. So is any point on the vertical line that "bisects" the dipole. In fact, if you extend the vertical line into and out of the screen, every point on that plane has the same potential, \(V=0\) (or \(V=V_\infty\) if we don't specify V_\infty).

A surface where the potential is the same at every point is called an equipotential surface or simply an equipotential; in two dimensions it can be referred to as an equipotential line, and is usually given a label like "0V".

Now suppose we move a target charge \(q_t\) along an equipotential surface from one point to another. What work does the electric field due on the charge during this motion? Well, the change in the system's potential energy is $$\Delta P=q_t\Delta V=q_t(0\u{V})=0\u{J}$$ so the electric force does no work on the charge. If you recall our discussion of work, force does no work on an object if the force is perpendicular to the motion. Therefore the electric force is perpendicular to the equipotential surface, and because the electric field points in the same or opposite direction as the electric force, the electric field is perpendicular to the equipotential surfaces.

That's worth repeating:
Electric field lines are perpendicular to equipotential surfaces, and vice versa.

That means that if we have an electric field line diagram, we can reconstruct the general shape of its equipotential surfaces. For instance, the electric field lines of a positive charge point away from the charge, radially. The equipotentials are surfaces that are perpendicular to radii: that is, spheres (or circles in two dimensions). The positive charge is a "peak" (a point of maximum potential), so the closest equipotential to it has the highest potential. The equipotentials of a dipole are more complicated, of course.

The figure shows a set of equipotentials (in black) and electric field lines (in blue). What direction should the electric field lines point?
The electric field points downhill, towards lower potential, and so the electric field lines must point to the right.