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fa02em_sol - PHY6938 Proficiency Exam Fall 2002 E&M 1 A...

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PHY6938 Proficiency Exam Fall 2002 September 13, 2002 E & M 1. A positive charge is uniformly distributed throughout a very long cylin- drical volume of radius R . The charge per unit volume is ρ . (a) Find the electric field ~ E everywhere as a function of the distance r from the axis of the cylinder. Let’s start by finding the electric field outside of the cylinder. Build a Gaussian surface out of a cylinder of length L and radius r > R . Then by symmetry the field must point radially outward on the curved part of the Gaussian surface and must be parallel to the surface on the end caps. This means that the flux of the electric field through this surface is I E · d A = E (2 πrL ) . (1) Now Gauss’s law tells us that this is also Q encl / 0 , where Q encl is the enclosed charge. The volume of the charged cylinder cut out by our surface is V = πR 2 L , so the enclosed charge is Q encl = πR 2 (2) and Gauss’s law gives E (2 πrL ) = 1 0 πR 2 Lρ, E = ρR 2 2 0 r . (3) Since the electric field points radially outward, we have E = ρR 2 2 0 r ˆ r . (4)
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