# 2.15 Hollow Spherical Shell

2.15 A hollow spherical shell carries charge density $$\displaystyle \mu = \frac{k}{\rho^2}$$ in a region $$r \le \rho \le R$$ . Find the electric field in the three regions: $$\rho < r$$ , $$r \le \rho \le R$$ , $$\rho > R$$ .

$$Q_{enc} = \int_r^R \int_0^{2 \pi} \int_0^{\pi} \mu \rho^2 \sin \phi \, d\theta \, d\phi \, d\rho = 4 \pi \int_r^R k \, d\rho = 4 k \pi ( R – r )$$

if $$\rho < r$$ then $$\left| \left| \vec{E} \right| \right| = 0$$ if $$r \le \rho \le R$$ $$\left| \left| \vec{E} \right| \right| ( 4 \pi \rho^2 ) = \frac{ 4 k \pi ( \rho - r ) }{ \epsilon_0 }$$ $$\left| \left| \vec{E} \right| \right| = \frac{ k ( \rho - r ) }{ \rho^2 \epsilon_0 }$$ If $$\rho > R$$

$$\left| \left| \vec{E} \right| \right| = \frac{ k ( R – r ) }{ \rho^2 \epsilon_0 }$$