What is the electric field and potential outside a spherical capacitor?

Question

Two concentric spheres form a spherical capacitor with the same charges (but opposite signal). I know, by Gauss's law, that the electric field must be zero (actually, the flux must be zero, but I can't see how the flux can be zero and the electric field is not zero). But why there is no net field outside the spherical capacitor if the negative charges in the (for example) external sphere create a electric field in every direction - including de direction pointing to outside the sphere? (in this case, there is no other electric that could cancel it).

So I think it's supposed to be like in this picture - and the external electric is falling off with the distance, as any electric field, and the electric field of the charge inside makes it fall off faster - though it is never zero.

If it is so, why does Gauss's law says it is zero? If it's not, why? (in this latter case, how can we explain it NOT using Gauss's law, using only the charges and electric fields?) I suppose the same holds for cylindrical capacitors

Answer 1

Alfred Centauri, yes I did [consider using superposition] and since the points outside the external sphere are closer to the the external sphere than the inside sphere, the "negative electric fiel" (electric field of the external sphere) is stronger than the "positive field" in the points outside the sphere. So the fields have the opposite directions and at first they could cancel each other but they don't because their magnitudes are different.

In the exterior region of an uniformly charged sphere, centered on the origin and with total charge Q, the electric field is identical to that of a point charge Q located at the origin.

Thus, by superposition, the electric field in the region outside of the concentric, uniformly charged spheres is simply the electric field due a point charge at the origin with charge equal to the sum of the total charge on each sphere.

In the case that the spheres have equal and opposite total charge, the sum is zero and, thus, the electric field in the exterior region is zero.

Answer 2

(actually, the flux must be zero, but I can't see how the flux can be zero and the electric field is not zero)

It seems like no one addressed your parentheses or at least not in a clear manner. I think my simple drawing will resolve your thoughts about how the flux can be zero without the electric field being zero. Note that flux can be thought of the net amount of flow through a given closed surface, with that being said there can obviously be a net flow of zero (since the same amount that comes in comes out) but there would still be a flow, in other words there would still be a non-zero electric field.

Note that this clears the common misconception of $\oint \vec E \cdot \vec {dl}=\frac{Q}{\epsilon_0}=0 \Rightarrow E=0$ which can be easily seen in the picture that it is $\bf{not}$ true.

Answer 3

So I think it's supposed to be like in this picture - and the external electric is falling off with the distance, as any electric field, and the electric field of the charge inside makes it fall off faster - though it is never zero.

The electric field for a sphere is: $E=\frac{kQ}{R}$. The inner sphere has a smaller $R$ than the larger sphere, but $E$ is larger for the inner sphere. The inner sphere does cancel the weaker $E$ from the larger sphere outside the larger sphere.