Why is it not possible for equipotential surfaces to intersect in this case? [duplicate]

Question

My textbook reads “ Two equipotential surfaces can never intersect because if they did, at the point of intersection, the field would have to have two directions (perpendicular to each surface) which is clearly absurd..”

I understand the fact that if there were to be a non-zero field at each point of the surface then since the surface is equipotential the Field must be perpendicular to the surface at each point.

But, is it also not possible that the field be zero throughout the two surfaces somehow?

In that case, considering the fact that the surfaces must have the same potential . According to me, the argument given in my textbook does not refute such a case.

Is it possible?

Thanks in advance!

Answer 1

In the case where the electric field at every point on two intersecting equipotential surfaces is zero, both of those equipotential surfaces are considered a single equipotential surface rather than two, since the potential at all the points on both the surfaces is the same. The potential on both the surfaces is the same because the work done in moving a unit charge from one point to another on the combined equipotential surface is zero since the electric field is zero.

So, no. This isn't a case where two equipotential surfaces intersect because there aren't two equipotential surfaces at all. It's just a single surface.