Question

Two metal spheres of radii `a’ and `b’ are far apart but are connected by a wire. Their combine charge is Q. Determine the charge on each of the conductor and also shows that the potential on each of the conductor is V= Q / (a+b).

Answer 1

Answer on Question #43978, Physics, Electrodynamics

Two metal spheres of radii ‘a and ‘b are far apart but are connected by a wire. Their combine charge is Q. Determine the charge on each of the conductor and also shows that the potential on each of the conductor is V= Q / (a+b).

Solution

A potential of sphere is given by

$V=k\frac{Q}{r}$

where $r$ is its radius. We know, that potential on each sphere must be equal each to other, because if not, the current will start to flow. Hence, we have equations

$V_{a}=k\frac{Q_{a}}{a}=V_{b}=k\frac{Q_{b}}{b}$

$Q_{a}+Q_{b}=Q$

From this we find that

$Q_{a}=\frac{Q}{1+b/a},\qquad Q_{b}=\frac{Q}{1+a/b}$

Potential is

$V_{a}=V=k\frac{Q_{a}}{a}=k\frac{Q}{a(1+a/b)}=k\frac{Q}{a+b}$