2.16 Coaxial Cable

2.16 A long coaxial cable carries a uniform volume density $$\mu$$ on the inner cylinder ( radius r ), and a uniform \textit{surface} charge density on the outer cylindrical shell ( radius R ). This surface charge is negative and of just the right magnitude so that the cable as a whole is electrically neutral. Find the electric field in each of the three regions:

for $$\rho < r$$ $$Q_{enc} = \int_0^{\rho} \int_0^l \int_0^{2 \pi} \mu r \, d\theta \, dl \, dr = 2 \mu \pi l \int_0^{\rho} r \, dr = \mu \pi l \rho^2$$ $$\left| \left| \vec{E} \right| \right| ( 2 \pi \rho l ) = \frac{ \mu \pi \rho^2 l }{ \epsilon_0 }$$ $$\left| \left| \vec{E} \right| \right| = \frac{ \mu \rho }{ 2 \epsilon_0 }$$ for $$r \le \rho \le R$$ $$\left| \left| \vec{E} \right| \right| ( 2 \pi \rho l ) = \frac{ \mu \pi r^2 l }{ \epsilon_0 }$$ $$\left| \left| \vec{E} \right| \right| = \frac{ \mu r^2 }{ 2 \rho \epsilon_0 }$$ for $$\rho > R$$ field is zero since electrically neutral.