Question:
Is it probable for the electric field to be zero in a region of space where the electric potential is not zero?
Answer:
Only if the electric potential has one and the same value throughout a region of space
, , and are all zero (and these are E_{x}, E_{y}, E_{z} respectively) so E is zero. Consider for instance a solid perfectly-conducting sphere at an electric potential of 125 volts.
We know that the electric field all over inside the sphere is zero. Additionally the value of 125 volts is the value of electric potential at every point in and on the sphere. Therefore the space occupied by a solid conducting sphere at an electric potential of 125 volts represents an example of a case in which the electric potential is not zero while the electric field is zero.