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 \( \rho > R \)

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