Question 24.98: In Fig. 24-71, a metal sphere with charge q = 5.00 μC and ra......
In Fig. 24-71, a metal sphere with charge q = 5.00 μC and radius r = 3.00 cm is concentric with a larger metal sphere with charge Q = 15.0 μC and radius R = 6.00 cm. (a) What is the potential difference between the spheres? If we connect the spheres with a wire, what then is the charge on (b) the smaller sphere and (c) the larger sphere?
(a) The potential difference is
\begin{aligned} \Delta V & =\frac{1}{4 \pi \varepsilon_0} \frac{Q}{R}-\frac{1}{4 \pi \varepsilon_0} \frac{q}{r}=\left(8.99 \times 10^9\, N \cdot m ^2 / C ^2\right)\left(\frac{15 \times 10^{-6} \,C }{0.060 \,m }-\frac{5.0 \times 10^{-6} C }{0.030 \,m }\right) \\ & =7.49 \times 10^5 \,V . \end{aligned}
(b) By connecting the two metal spheres with a wire, we now have one conductor, and any excess charge must reside on the surface of the conductor. Therefore, the charge on the small sphere is zero.
(c) Since all the charges reside on the surface of the large sphere, we have
Q^{\prime}=Q+q=15.0 \,\mu C +5.00\, \mu C =20.0 \,\mu C .