Chapter 4, Problem 8P

An electron beam shaped like a circular cylinder of radius r₀ carries a charge density given by

${\rho }_v=\left(\frac{-{\rho }_0}{1+r^2}\right)({\text{C/m}}^{\text{3}})$

where ρ₀ is a positive constant and the beam’s axis is coincident the z axis.

1. (a) Determine the total charge contained in length L of the beam.
2. (b) If the electrons are moving in the +z direction with uniform speed u, determine the magnitude and direction of the current crossing the z-plane.