Question

What is the capacitance of concentric spherical capacitors of radius a(inner) and b? Inner is earthed and outer is positively charged. Derive.

Answer 1

This may be the result of the case when the outer one is earthed. But here the inner is earthed. I have found that this arrangement forms parallel combination of two capacitors. But I am unknown how.The result given is 4\[\pi \epsilon\] b^2/(b-a)

Answer 2

\[\int\limits_a^b\vec{E}_1\cdot d\vec{r}+\int\limits_b^{+\infty}\vec{E}_2\cdot d\vec{r}=0\]\[\frac{Q_{IN}}{4\pi\varepsilon}\int\limits_a^b\frac{dr}{r^2}+\frac{Q_{IN}+Q_{OUT}}{4\pi\varepsilon}\int\limits_b^{+\infty}\frac{dr}{r^2}=0\]\[\frac{Q_{IN}}{4\pi\varepsilon}\left(\frac{1}{a}-\frac{1}{b}\right)+\frac{Q_{IN}+Q_{OUT}}{4\pi\varepsilon}\frac{1}{b}=0\]\[\frac{Q_{IN}}{4\pi\varepsilon}\frac{1}{a}+\frac{Q_{OUT}}{4\pi\varepsilon}\frac{1}{b}=0\quad\Rightarrow\quad Q_{IN}=-\frac{a}{b}Q_{OUT}\]\[U=\frac{Q_{IN}+Q_{OUT}}{4\pi\varepsilon}\frac{1}{b}=\frac{-a/b+1}{4\pi\varepsilon}\frac{1}{b}Q_{OUT}\]\[C=\frac{Q_{OUT}}{U}=4\pi\varepsilon\frac{b}{1-a/b}=4\pi\varepsilon\frac{b^2}{b-a}\]