Source: High school physics (Chinese)
Problem Sets:
Problem
A point charge $q$ is placed at the center of an uncharged metal spherical shell.
- What is the field strength at point $A$ outside the shell (distance $r$ from the center) and at point $B$ inside the cavity (distance $r'$ from the center, between $q$ and the inner surface)?
- If $q$ is displaced from the center, do the field strengths at $A$ and $B$ change?
- What measure should be taken so that the shell shields $q$'s influence from the outside world?
(1) $E_A = \dfrac{kq}{r^2}$ (radially outward); $E_B = \dfrac{kq}{r'^2}$ (radially outward from $q$). (2) $E_A$ unchanged; $E_B$ changes. (3) Ground the metal shell.
By induction the inner surface of the shell carries $-q$ and the outer surface carries $+q$.
At $A$ (outside): a Gaussian sphere of radius $r$ encloses net charge $q$, so $E_A = \dfrac{kq}{r^2}$, directed radially outward.
At $B$ (inside the cavity, with $q$ centered): the inner-surface charge $-q$ is uniformly distributed and contributes zero field inside the cavity by spherical symmetry. Only the central charge $q$ contributes: $E_B = \dfrac{kq}{r'^2}$, directed radially outward from $q$.
If $q$ is displaced from the center: the outer surface still carries $+q$ uniformly distributed (the conductor screens the asymmetry from the outside), so $E_A$ is unchanged. Inside the cavity, the inner-surface induction is no longer uniform, so $E_B$ changes in both magnitude and direction.
To shield $q$'s influence from the outside world: ground the shell. The $+q$ on the outer surface flows to ground, leaving zero outer-surface charge and thus zero field outside.