Question

 Two concentric conducting spheres are filled with an insulating material of uniform charge density ρ0. The inner sphere has radius ra and is held at a potential φ0. The outer sphere has a radius rb and is grounded. (a) Determine the potential and field everywhere in space. (b) What is the capacitance of this object?

Answer 1

Given,

charged density of the insulating material $=\rho_o$

The radius of inner sphere $=r_a$

Potential $=\phi_o$

Radius of outer sphere $=(r_b)$

$V_{in}= \frac{dq}{4\pi \epsilon_o r_a}$

Net potential difference, if outer surface was not grounded

$dq = \rho_o4\pi r_a^2 dr$

$V= \int_{r_a}^{r_b}\frac{4\pi r^2\times \rho_o}{4\pi \epsilon_o r}dr$

$\Rightarrow V = \int_{r_a}^{r_b}\frac{\rho_o}{\epsilon_o}rdr$

$\Rightarrow V =\frac{\rho_o}{2\epsilon_o}(r_b^2-r_a^2)$

Capacitance $(c)=\frac{Q}{V}$

net charge$(dq)=\rho_o 4\pi r^2 dr$

Now,integrating both side,

$\int_0^Q dq=\int_{ra}^{r_b}\rho_o 4\pi r^2 dr$

$Q=\dfrac{4\pi \rho_o}{3}(r_b^3-r_a^3)$

Hence the required capacitance

$C=\frac{\dfrac{4\pi \rho_o}{3}(r_b^3-r_a^3)}{\dfrac{\rho_o}{2\epsilon_o}(r_b^2-r_a^2)}$

$\Rightarrow C= \dfrac{8\pi \epsilon_o}{3}\dfrac{(r_b-r_a)(r_b^2+r_a^2+r_a r_b)}{(r_b-r_a)(r_b+r_a)}$

$\Rightarrow C = \dfrac{8 \pi \epsilon_o(r_b^2+r_a^2+r_a r_b)}{(r_b+r_a)}$