Question

Two spheres have radii a and b and their centres are at a distance d apart. The capacitance of this system is;
$C=\frac{4\pi {\epsilon }_o}{\frac{1}{a}+\frac{1}{b}\pm \frac{2}{d}}$
provided that d is large compared with $a$ and $b$ .

Show that as $d$ approaches infinity the above result reduces to that of two isolated spheres in series.

Answer 1

If $d\to \infty$,

Then $C=\frac{4\pi {\epsilon }_o}{\frac{1}{a}+\frac{1}{b}}$

$=\frac{\left(4\pi {\epsilon }_o\right)\left(ab\right)}{a+b}$
In series,
$C_{net}=\frac{C_1{C}_2}{C_1+C_2}=\frac{\left(4\pi {\epsilon }_{o}a\right)\left(4\pi {\epsilon }_{o}b\right)}{\left(4\pi {\epsilon }_{o}a\right)+\left(4\pi {\epsilon }_{o}b\right)}$
$=\frac{\left(4\pi {\epsilon }_o\right)ab}{a+b}$