HW05Sol - σ 2 σ 1 and their relative signs be for the...

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Jim Guinn’s PHYS2212 Assignment #5 Solutions 1. Consider two infinitely long, concentric cylindrical shells. The inner shell has a radius R 1 and carries a uniform surface charge density σ 1 , and the outer shell has a radius R 2 and carries a uniform surface charge density σ 2 . Note that due to the symmetry of the problem the electric field is always radial with respect to the axis of the cylinders. a) Use Gauss’s Law to find the electric field in the regions r < R 1 , R 1 < r < R 2 , and r > R 2 . We choose a cylinder of length L and radius r, concentric with the charged cylinders. For r < R 1 , the charge enclosed is zero, so E = 0 for r < R 1 . For R 1 < r < R 2 , we have that Φ = EA = E (2 π r L) = Q enc / ε o = σ 1 (2 π R 1 L) / ε o , so E = σ 1 R 1 / ( ε o r) for R 1 < r < R 2 . For r > R 2 we have that Q enc / ε o = ( σ 1 2 π R 1 L + σ 2 2 π R 2 L) / ε o , so E = ( σ 1 R 1 + σ 2 R 2 ) / ( ε o r) for r > R 2 . b) What should the ratio of the surface charge densities
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Unformatted text preview: σ 2 / σ 1 and their relative signs be for the electric field to be zero at r > R 2 ? To have E = 0 for r > R 2 , we must have σ 1 R 1 + σ 2 R 2 = 0 , or σ 2 / σ 1 = - R 1 /R 2 . The charge densities must have opposite signs and the inverse ratio of their respective radii. 2. Starting from Gauss’s Law, derive the electric field for a point charge, Q, i.e. Coulomb’s Law. Look at your class notes. 3. Starting from Gauss’s Law, derive the electric field for an infinite line charge with uniform linear charge density, λ . Look at your class notes. 4. Starting from Gauss’s Law, derive the electric field for an infinite sheet of charge with uniform surface charge density, σ . Look at your class notes....
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This note was uploaded on 01/23/2012 for the course PYSICS 2212 taught by Professor Jimguinn during the Fall '11 term at Georgia Perimeter.

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