Question

A spherical capacitor is made of two conducting spherical shells of radii $p=5\text{cm}$ and $q=15\text{cm}$. The space between the shells is filled with a dielectric of dielectric constant $K=10$ upto a radius $r=9\text{cm}$ as shown in the figure. Calculate the capacitance (in $\text{pF}$). [Nearest integer value]

Answer 1

We know that, Capacitance of a spherical capacitor is given by $$C=\frac{4\pi {\epsilon }_{0}ab}{a-b}$$From the given figure, it is clear that capacitors $C_{pq}$ and $C_{qr}$ are in series.


Capcitance of spherical capacitor $pq$ , $C_{pq}=\frac{K4\pi {\epsilon }_{0}pq}{(p-q)}$

Capcitance of spherical capacitor $qr$ , $C_{qr}=\frac{4\pi {\epsilon }_{0}qr}{(q-r)}$

Thus, equivalent capacitance $\frac{1}{C_{eq}}=\frac{1}{C_{pq}}+\frac{1}{C_{qr}}$

$\Rightarrow \frac{1}{C_{eq}}=\frac{(p-q)}{K4\pi {\epsilon }_{0}pq}+\frac{(q-r)}{4\pi {\epsilon }_{0}qr}$

$\Rightarrow C_{eq}=\frac{K4\pi {\epsilon }_{0}pqr}{Kp(q-r)+q(r-p)}$

Substituting the data given in the question we get,

$C_{eq}=\frac{1}{9\times {10}^9}\times \frac{10\times 15\times 5\times 9\times {10}^{-2}}{[10\times 5(15-9\left)\right]+\left[15\right(9-5\left)\right]}$

$C_{eq}=\frac{750}{36}\times {10}^{-12}\approx 21\text{pF}$

Accepted answer : $21$