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August 11, 2017

Capacitance of a Spherical Capacitor

Try this problem, useful for Physics Olympiad, based on Capacitance of a Spherical Capacitor.

The Problem:

A spherical capacitor has inner radius (a) and outer radius (b). It is filled with an inhomogeneous dielectric with permittivity $$ \epsilon=\epsilon_0K/r^2$$ for (a<r<b). The outer sphere is grounded and a charge is placed on the inner sphere. Find the capacitance of the system.

Solution:
The electric field at any inside point is $$\vec{E}=\frac{Q}{4\pi\epsilon r^2}\hat{r} =\frac{Q}{4\pi\epsilon_0K}\hat{r}$$ where (Q) is the charge on the inner sphere.
Now, the potential difference between the spheres is $$ V=-\int_{b}^{a}
\vec{E}.\vec{dr}$$$$=\int_{a}^{b}\frac{Q}{4\pi\epsilon_0K}dr$$$$=\frac{Q}{4\pi\epsilon_0K}(b-a)
$$
Capacitance $$ C=\frac{Q}{V}=\frac{4\pi\epsilon_0K}{b-a}$$

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