Source: High school physics (Chinese)
Problem Sets:
Problem
Two concentric spherical conducting shells: the inner shell carries charge $Q$, and the outer shell has zero net charge.
- What is the charge $Q_1$ on the inner surface of the outer shell, and the charge $Q_2$ on its outer surface?
- At a point $P$ outside the outer shell at distance $r$ from the center, what is the total electric field?
- What field at $P$ is produced by $Q_2$ alone? Does $Q$ produce a field at $P$? Does $Q_1$? What changes if the outer shell is grounded?
(1) $Q_1 = -Q$ (inner surface of outer shell); $Q_2 = +Q$ (outer surface of outer shell). (2) $E_{\text{total}} = kQ/r^2$. (3) All three charges ($Q$, $Q_1$, $Q_2$) produce fields at $P$ individually; only their sum equals $kQ/r^2$. When the outer shell is grounded, $Q_2 \to 0$ and the field outside the shell becomes zero (electrostatic shielding).
(1) Inside the conductor of the outer shell, $\mathbf{E} = 0$. Applying Gauss's law to a Gaussian surface within the outer shell's conductor, the enclosed charge must be zero, so the inner surface of the outer shell carries $Q_1 = -Q$. The outer shell has zero net charge, so its outer surface carries $Q_2 = Q$.
(2) Apply Gauss's law to a sphere of radius $r > R_{\text{outer}}$: total enclosed charge is $Q + Q_1 + Q_2 = Q$. So
$$E_{\text{total}} = \dfrac{kQ}{r^2}, \quad \text{radially outward}.$$(3) Each charge distribution contributes (by superposition and the shell theorem):
$$E_{Q_2}(P) = \dfrac{kQ}{r^2} \text{ outward}, \quad E_{Q}(P) = \dfrac{kQ}{r^2} \text{ outward}, \quad E_{Q_1}(P) = -\dfrac{kQ}{r^2} \text{ (inward)}.$$All three contribute; their sum at $P$ is $\dfrac{kQ}{r^2}$ outward, matching (2).
If the outer shell is grounded, $Q_2$ flows to ground. Only $Q$ (at center) and $Q_1 = -Q$ remain. The total enclosed charge for any $r > R_{\text{outer}}$ is now $0$, so the field outside vanishes: $E_{\text{outside}} = 0$. The field between the two shells is unchanged ($E = kQ/r^2$, due to the central charge only). The outer shell now shields the outside world from the inner charge.