The electric field intensity and the electric potential are $E$ and $V$ respectively. Which of the following is correct?

Solution:

The relationship between electric field intensity ($E$) and electric potential ($V$) is given by $E = -\nabla V$. This means the electric field is the negative gradient of the electric potential.

Let's analyze each option:

(1) If $E \neq 0, V$ cannot be zero:

This statement is incorrect. Consider an electric dipole. At any point on its equatorial plane, the electric potential $V = 0$, but the electric field $E$ is non-zero and points opposite to the dipole moment. Thus, it is possible for $V$ to be zero even if $E \neq 0$.

(2) If $V \neq 0, E$ cannot be zero:

This statement is incorrect. Consider a uniformly charged spherical shell. Inside the shell, the electric field $E = 0$, but the electric potential $V$ is constant and equal to the potential on the surface of the shell, which is generally non-zero. Thus, it is possible for $E$ to be zero even if $V \neq 0$.

(3) If $V = 0, E$ must be zero:

This statement is incorrect. As explained in (1), for an electric dipole, $V=0$ on the equatorial plane, but $E \neq 0$.

(4) If $E = 0, V$ may not be zero:

This statement is correct. As explained in (2), inside a uniformly charged spherical shell (or any conductor in electrostatic equilibrium), the electric field $E = 0$ everywhere, but the electric potential $V$ is constant throughout the volume and on the surface, and this constant potential is generally non-zero. For example, if a conductor is charged, its potential will be non-zero, but the field inside is zero.

Therefore, the correct statement is that if $E = 0$, $V$ may not be zero.