Electrostatics: Conceptual Doubts Cleared (8)

Question

Inside a hollow conducting sphere of radius $R$ and total charge $Q$, a small point charge $q$ is placed at the centre. Find (a) the electric field at a point inside the cavity at distance $r < R$ from the centre, and (b) the field outside the sphere at distance $d > R$. Does the answer change if $q$ is moved off-centre inside the cavity?

Solution — Step by Step

Final answers: $E_{\text{inside}} = kq/r^2$, $E_{\text{outside}} = k(q+Q)/d^2$. Outside field is unchanged when $q$ shifts.

Why This Works

A conductor has free electrons that rearrange to make the field inside the conductor’s metal exactly zero. To do that, the inner surface acquires an induced charge of $-q$, and (since the conductor has total charge $Q$) the outer surface carries $Q + q$.

The outer surface charge always distributes uniformly when there is no external field, regardless of where $q$ sits inside. That uniformity is why the outside field looks like it comes from a point charge $(q+Q)$ at the centre.

Alternative Method

Use the principle of charge induction explicitly. Inner surface: $-q$ (cancels the field of $q$ inside the metal). Outer surface: $Q + q$ (conservation of charge on the conductor). Apply Gauss’s law to either surface — same answers.

Common Mistake

Writing the inside field as $kq/r^2$ when $q$ is off-centre — but using $r$ measured from the geometric centre of the sphere instead of from $q$. The correct $r$ is the distance from $q$ itself. The conductor does not change that — it only screens the field beyond its outer surface.