- #1

cepheid

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**Problem**: A long coaxial cable carries a uniform

*volume*charge density [itex] \rho [/itex] on the inner cylinder (radius [itex] a [/itex]), and a uniform

*surface*charge density on the outer cylindrical shell (radius [itex] b [/itex]). The surface charge is negative and of just the right magnitude so that the cable as a whole is electrically neutral. Find the electric field in each of the three regions: (i) inside the inner cylinder (s < a), (ii) in between the cylinders (a < s < b), (iii) outside the cable (s > b).

My thoughts: First of all, from the statement that the volume charge distribution on the inner cylinder was equal and opposite to the surface charge distribution on the outer one, I extracted this relation, hoping it would come in useful:

[tex] \iiint_{V_{\text{inner}}} {(\rho dV)} = - \iint_{S_\text{outer}}{(\sigma dS)} [/tex]

Correct?

My next thought was that the electric field in region iii (outside the cable) should be zero everywhere. My reasoning was that any Gaussian surface enclosing a section of the cable encloses

*zero*net charge. So it would not make any sense for there to be an electric field at any point on said surface.

My instinct tells me that by symmetry, the contribution to the electric field due to the outer cylinder in region i (inside the inner one) cancels itself out completely, but I'm not sure how to prove it.

I'm a bit new to this type of problem, so I appreciate hearing whether I'm right so far, and how to go about solving the rest of the problem.