I have a conducting sphere which has a cavity in it. The cavity is not at the center of the sphere. If a charge $+q$ is placed inside the cavity (with the sphere remaining neutral as a whole), what would be the electric field at a point outside the sphere?
Would the answer change if the charge is moved to a different location within the cavity?
Would the answer depend on the location of the cavity?
Best Answer
The field would, strangely enough, be equal to that of a uniformly charged sphere of charge $q$. The placement and shape of the cavity doesn't change the field outside the sphere.
By using Gauss's law, we see that $$\oint \mathbf E \cdot d \mathbf a = \frac{Q_{enc}}{\epsilon _0}$$
Where $Q_{enc} = q -q +q = q$
The point charge in the cavity will induce an equal and opposite charge on the cavity wall, canceling out the contribution of the point charge outside the sphere, while the sphere must now take on a uniformly distributed charge of $q$, since the sphere is to remain neutral.
$\mathbf E$ inside the cavity depends on the geometry and position of the charge.