A long cylindrical rod of radius R has a charge density of ρ = p(r) that is not constant, but only depends on the distance from the rod center. Show that the radial electric field component obeys d(rE)/dr=ρr/ϵ.
(Hint: consider the flux through an annular region bounded by r and r + dr )

Question

A long cylindrical rod of radius R has a charge density of ρ = p(r) that is not constant, but only depends on the distance from the rod center. Show that the radial electric field component obeys d(rE)/dr=ρr/ϵ.
 (Hint: consider the flux through an annular region bounded by r and r + dr )

anonymous · Answer

Radial electric field an be obtained by using a Gaussian surface, a cylinder within the rod and for which the integral of E times the surface area perpendicular to E equals the enclosed charge divide by epsilon0. Larger r has more enclosed charge but more area.

anonymous · Answer

2 pi r L E = L (1/epsilon0) integral rho(r) 2 pi r dr

where rho(r) is the charge density as a function of distance from the axis of the cylinder