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 ?
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1$\begingroup$ What is your notion of "region" here? For "the interior of a set with non-zero 3d volume", the statement is correct. For other notions, as you point out, not. I'm unsure what the physics question is. $\endgroup$– ACuriousMind ♦Commented Jan 19, 2022 at 17:35
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$\begingroup$ The question is exactly as stated in the title of the post. I've been given four options : 1) The electric field is uniform 2) The electric field is zero 3) There can be no charge inside the region 4) Both 2 and 3. The answer according to the source is (4) . But even option (1) is correct according to the logic I used above. I feel like the answer given at the end of the problem set doesn't consider the fact potential can be constant on an equipotential surface with a field that has a non zero value perpendicular to the surface of the equipotential surface. $\endgroup$– Noah CygnusCommented Jan 19, 2022 at 17:58
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$\begingroup$ '[...] potential can be constant on an equipotential surface with a field that has a non zero value perpendicular to the surface of the equipotential surface. [...]" But your problem is about a field where the potential is the same EVERYWHERE, not just on a set of equipotential surfaces. If a test charge goes from any point A to any point B no work is done on it by the field, as its potential energy doesn't change. This can be the case only if the field strength is zero everywhere. $\endgroup$– Philip WoodCommented Jan 19, 2022 at 18:21
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$\begingroup$ I guess I'm having a hard time understanding the question. So basically what's being asked here is if potential is constant everywhere in space what is the value of the field? $\endgroup$– Noah CygnusCommented Jan 19, 2022 at 18:31
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$\begingroup$ What do you use as the definition of potential and electric field? Do you define $\vec{E}=-\nabla V$ or perhaps in a non calculus manner as $E=-\frac{\Delta V}{\Delta s}$ (s is some path through a field)? $\endgroup$– TriatticusCommented Jan 19, 2022 at 18:43
2 Answers
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.
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.
