Two connected conducting spheres

Problem

A conducting sphere of radius $r_1 = 5\;\text{cm}$ carries charge $Q_1 = +8{,}0\;\mu\text{C}$. A second sphere of radius $r_2 = 3\;\text{cm}$ carries charge $Q_2 = -2{,}0\;\mu\text{C}$. The two spheres are connected by a thin, long conducting wire (the distance between the centres is much greater than the radii). Find the charge on each sphere after the connection.

Solution

Setup. After the connection, the two spheres and the wire form a single conductor in electrostatic equilibrium. Two constraints hold: conservation of total charge and equality of the potentials.

Constraint 1 — conservation of charge. The total charge cannot disappear or be created: it only redistributes. $Q_1' + Q_2' = Q_1 + Q_2 = 8{,}0 + (-2{,}0) = +6{,}0\;\mu\text{C}$

Constraint 2 — equal potential. In a conductor at equilibrium the potential is the same everywhere. Since the distance between the spheres is much greater than the radii, each sphere behaves as an isolated point charge, with surface potential $V = kQ/r$: $V_1' = k\,\frac{Q_1'}{r_1} = k\,\frac{Q_2'}{r_2} = V_2' \;\Rightarrow\; \frac{Q_1'}{Q_2'} = \frac{r_1}{r_2} = \frac{5}{3}$

Solving the system. From $Q_1' = \tfrac{5}{3}Q_2'$ substituted into the first constraint: $\begin{aligned} \tfrac{5}{3}\,Q_2' + Q_2' &= 6{,}0\;\mu\text{C} \\ \tfrac{8}{3}\,Q_2' &= 6{,}0\;\mu\text{C} \\ Q_2' &= \frac{6{,}0 \times 3}{8} = 2{,}25\;\mu\text{C} \end{aligned}$ $Q_1' = 6{,}0 - 2{,}25 = 3{,}75\;\mu\text{C}$

Check. The two potentials coincide: $V_1' = (9\cdot 10^9)\,\frac{3{,}75\cdot 10^{-6}}{0{,}05} = 6{,}75\cdot 10^{5}\;\text{V} \qquad V_2' = (9\cdot 10^9)\,\frac{2{,}25\cdot 10^{-6}}{0{,}03} = 6{,}75\cdot 10^{5}\;\text{V}\;\checkmark$

$\ev{Q_1' = 3{,}75\;\mu\text{C}, \qquad Q_2' = 2{,}25\;\mu\text{C}}$

Remark. The smaller sphere collects less charge, but its surface density $\sigma_2 = Q_2'/(4\pi r_2^2)$ is greater than $\sigma_1$: the external field is more intense near the smaller sphere. In general, conductors with smaller radii of curvature (points) concentrate the field, the principle on which the lightning rod is based.

Links

Topics: Electric field and potential Concepts: Conductors in electrostatic equilibrium · Electric potential · Electric charge Objects: Charged sphere