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2-1 soln - Question 1 A very long(you can count it as...

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Question 1 A very long (you can count it as infinite) ceramic (i.e. nonconducting) cylindrical pipe of inner radius a and outer radius b has accidentally become charged with a uniform charge density ρ . You are concerned that the electric field due to this charge may be affecting sensitive instruments inside the inner radius of the pipe, embedded in the pipe, and hanging outside the pipe. Having learned Gauss’ Law you can calculate the fields that might be upsetting the instruments. Consider a length of the pipe L Inside the pipe The contained charge is 0. Therefore by Gauss’ Law, E=0 from r=0 to r=a Outside the pipe draw a Gaussian cylinder parallel to the pipe at a distance r Contained charge is ( ) 2 2 b a L π ρ By Gauss’ Law 0 q E dS ε = G G i v a b ρ L
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From symmetry E is perpendicular to the cylinder and constant at radius r, therefore 2 E dS E dS rLE π = = G G i v Then ( ) ( ) 2 2 0 2 2 0 2 2 b a L rLE b a E r π ρ π ε ρ ε = = This is the magnitude of E, the vector field is in the direction of r ˆ E Er = G
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