Answer to Question #174834 in Electricity and Magnetism for charles king

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

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

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

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