Question

If V = 0 at a point in space, must E ?= 0? If E ?= 0 at some point, must V = 0 at that point?
Explain. Give examples for each.
b. Can two equipotential lines cross? Explain.

Answer 1

A)

The charge of V over a very short distance is E, they are not explicitly dependent quantities. So both answer is no. For example In the center of a dipole V = 0 but E is finite. And the point midway between two positive charges. E is 0 there, but V is finite.

B)

The electric field on an equipotential line is the perpendicular on the tangent at that point. If, say, two equipotential lines cross then at the same point electric fields will have two different directions which is absurd. So two equipotential lines can't cross.