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Unformatted text preview: EE1 Spring 2008 Homework #4 Solutions 1) An electron beam shaped like a circular cylinder of radius r carries a charge density given by: ρ v = (- ρ 1 + r 2 )( C m 3 ) where ρ is a positive constant and the beam’s axis is coincident with the z-axis. a) Determine the total charge contained in the length L of the beam. This can be determined by evaluating the volume integral of the volume charge density with the volume of the cylinder. Z v ρ v dV =- πρ Lln (1 + r 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. Multiplying the charge density by the speed will result in the current density of the electron beam. J = ρ v u A simple surface integral can then determine the current I crossing the z-plane. I = Z J · d S I =- πuρ ln (1 + r 2 ) The current is traveling in the- ˆ z direction. 1 2) A coaxial conductor has radii a = 0.8 mm and b = 3 mm and a polystyrene dielectric for which r = 2 . 56. If P = 2 r ˆ r ( nC/m 2 ) in the dielectric, find: a) D and E as functions of r.a) D and E as functions of r....
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- Spring '08
- Electric charge, 1 cm, 1 g, 2 L, Wave Electromagnetics