Answer to Question #156062 in Physics for qudsia

Question

Question-5

(a)   If the electric field is zero throughout a region, must the electric potential also be zero in that region?

(b)  If two points are at the same potential, are there any electric field lines connecting them?

(c)   Is the electric potential necessarily constant over the surface of a conductor?

Answer 1

a) The electric field is the negative gradient of the electric potential:

$$\mathbf{E} = -\nabla\varphi$$

Thus, the electric potential does not have to be zero for the electric field to be zero. It is enough for the potential to be constant in this region: $\varphi = const$.

b) No, there are no electric field lines connecting them, for the same reason as in a). The gradient of constant is zero.

c) In any static charge distribution the electric field inside the conductor is zero. If it is not zero, charges move until they arrange such manner to maintain zero field inside. After that they stop, since there is no field acting on them any more.

Since the field is zero, the electric potential should be constant (as argued in a)) inside the conductor and on its surface as well.

Answer. a) No, b) No, c) Yes.