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A spherical capacitor has an inner sphere of radius $12 \mathrm{~cm}$ and an outer sphere of radius $13 \mathrm{~cm}$. The outer sphere is earthed and the inner sphere is given a charge of $2.5 \mu \mathrm{C}$. The space between the concentric spheres is filled with a liquid of dielectric constant 32 .
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with latter is much smaller.
(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with latter is much smaller.
Solution:
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Verified Answer
Given, $\mathrm{r}_1=12 \mathrm{~cm}=12 \times 10^{-2} \mathrm{~m}, \mathrm{r}_2=13 \mathrm{~cm}=13 \times 10^{-2} \mathrm{~m}$, $\mathrm{q}=2.5 \mu \mathrm{C}=2.5 \times 10^{-6} \mathrm{C}, \mathrm{K}=32$
(a) By formula, capacitance $\mathrm{C}$
$$
\begin{aligned}
&=K .4 \pi \varepsilon_0 \frac{r_2 r_1}{r_2-r_1} \\
&=\frac{32 \times 13 \times 10^{-2} \times 12 \times 10^{-2}}{9 \times 10^9\left(13 \times 10^{-2}-12 \times 10^{-2}\right)} \\
&=\frac{1664}{3} \times 10^{-11}=5.55 \times 10^{-9} \mathrm{~F} .
\end{aligned}
$$
(b) Potential of inner sphere, $V=q / C$ $=\frac{2.5 \times 10^{-6} \times 3}{1664 \times 10^{-11}}=4.5 \times 10^2 \mathrm{~V}$.
(c) Capacitance of sphere, $\mathrm{C}$
$$
=4 \pi \varepsilon_0=\frac{12 \times 10^{-2}}{9 \times 10^9}=1.33 \times 10^{-11} \mathrm{~F} \text {. }
$$
It is small because it forms no capacitor.
(a) By formula, capacitance $\mathrm{C}$
$$
\begin{aligned}
&=K .4 \pi \varepsilon_0 \frac{r_2 r_1}{r_2-r_1} \\
&=\frac{32 \times 13 \times 10^{-2} \times 12 \times 10^{-2}}{9 \times 10^9\left(13 \times 10^{-2}-12 \times 10^{-2}\right)} \\
&=\frac{1664}{3} \times 10^{-11}=5.55 \times 10^{-9} \mathrm{~F} .
\end{aligned}
$$
(b) Potential of inner sphere, $V=q / C$ $=\frac{2.5 \times 10^{-6} \times 3}{1664 \times 10^{-11}}=4.5 \times 10^2 \mathrm{~V}$.
(c) Capacitance of sphere, $\mathrm{C}$
$$
=4 \pi \varepsilon_0=\frac{12 \times 10^{-2}}{9 \times 10^9}=1.33 \times 10^{-11} \mathrm{~F} \text {. }
$$
It is small because it forms no capacitor.
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