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A long straight wire with linear charge density lambda runs along the axis of a thin hollow metallic cylindrical shell of radius R. The shell has a net linear charge density 2*lambda. Assume lambda > 0. Which of the following are correct?
- (A) E(r > R) = 3*lambda/(2*pi*epsilon0) * (r_hat/r)
- (B) E(r < R) = 3*lambda/(2*pi*epsilon0) * (r_hat/r)
- (C) Linear charge density on inner surface of cylinder is -lambda
- (D) Linear charge density on outer surface of cylinder is 3*lambda
Correct answer: (A) E(r > R) = 3*lambda/(2*pi*epsilon0) * (r_hat/r)
Solution
For r > R: total enclosed charge per unit length = lambda + 2*lambda = 3*lambda, so E = 3*lambda/(2*pi*epsilon0*r). For r < R (outside wire, inside conductor): field inside conductor = 0; Gauss's law implies inner surface has -lambda. Net shell charge = 2*lambda, outer surface = 2*lambda - (-lambda) = 3*lambda. Options A, C, D are correct; B is wrong (field for r < R comes only from wire = lambda/(2*pi*epsilon0*r), not 3*lambda).
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