What is the electric field in a region of constant potential? I am working on a problem set on electrostatic potential and there is this one particular question posed that states "The electric field in a region of constant potential is?". The answer given at the end of the problem set is "Electric field is zero and there can be no charge inside the region". That sounds so close minded to me like they are speaking of a particular case. My intuition says sure that answer is correct as well but a broader,  more generalized answer will be there can be an electric field in a region of constant potential but it has be uniform and perpendicular to the region. For example a hollow sphere that ENCLOSES a charge has a constant field on its surface given by $$E = \frac{KQ}{R^2}$$ where $R$ is the radius of the sphere and every point on the surface has the same potential. Am I wrong ?
 A: As pointed out in the comments, the word region is being used to refer to a set of points with non-zero 3D volume.  In contrast, your counterexample is a 2D surface, which is not a region in the sense meant by your instructor.

If I use the mathematical definition, and try to get the field by differentiating a constant potential the answer will be zero. But I'm trying to look at it through the lens of intuition not mathematics.

Intuition and mathematical precision are not mutually exclusive. Intuition is built from experience, and mathematical precision is a tool which you should use to refine it.
Indeed from a certain point of view, it is you who is pointing out a mathematical technicality.  You are making the point that if we consider a surface or a curve, it's possible for the potential along the surface or curve to be constant even in the presence of a non-zero electric field, which is certainly true. However, if the region in question is an open subset of $\mathbb R^3$ - which means that each point in the region is surrounded by a small 3D ball which is also in the region - then a constant potential means that the electric field must be identically zero everywhere within the region.
So the resolution to the apparent contradiction is simply defining your terms more clearly - that is, with more mathematical precision.
A: In electrostatics the component of the electric field with respect to a direction is proportional to the rate of change of the potential per unit length along this direction, what we call the gradient. A sphere with uniform surface charge density
with its empty interior is a equipotential region (constant potential). But on the surface and along any direction towards the exterior there exists a non-zero rate of change of the potential per unit length, that's why the non-zero electric field. In other terms the sphere and its surroundings is not a region of constant potential.
At any point on the (equipotential) spherical surface the tangential component of the electric field is zero, that's why the electric field  is perpendicular (but not zero) to the surface.
The potential is an everywhere continuous function but with discontinuous first derivative on the surface.
