If the intensity at a point in an electric field is zero, will the electric potential be also zero at that point? If the electric potential be zero at a point, will the intensity be also zero at the point?

Explanation

If the intensity of the electric field be E and potential be V, then the relation between them is, `E = (dV)/(dx)` So, if `E = 0` at any point, we have `(dV)/(dx) = 0` or, V = constant, Thus, the potential has a constant value, not necessarily zero, around that point. Now, for example, we consider an electric dipole (q,-q) placed at AB along the x-axis. The variation of the potential V with the distance x is shown in the figure. At the mid-point O, clearly, V = 0, but the slope of the V-x curve is not zero, i.e., `(dV)/(dx) != 0, "As" E = (dV)/(dx)`, we can conclude that `E != 0` at O.
Therefore, if the potential is zero at a point, but it is not a constant in the immediate neighbourhood of that point, the electric field intensity is non-zero there.
In particular, the potential due to a dipole at any point on the perpendicular bisector is zero, but the field at that point is certainly non -zero being directed opposite to the direction of the dipole moment.