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A long coaxial cable consists of an inner solid cylinder, radius a, and an outer thin coaxial cylindrical shell, radius b. The outer shell carries a uniform surface charge density σ.

Find the uniform volume charge density ρ that the inner cylinder must have in order that the whole cable (inner + outer) is neutral.

2. Homework Equations

Acylinder = 2πbl

Vcylinder = πa^2l

Qenc = ∫Vρdτ

3. The Attempt at a Solution

I started by using dq = sigma*dS, and integrating to get Q_b = -2(pi)(sigma)bL. That should be the total charge on the outer cylindrical shell.

Then I set -Q_b = Q_a, so the net charge of the entire system would be equal to 0.

Next I used Q_a = row*integral(dVolume) = -Q_b

to get to (row)(pi)(a^2)(L) = 2(pi)(sigma)(b)(L)

Solving for row gave me:

row = [2(sigma)(b)] / a^2

Does this look like a proper method and correct solution?

Thank you so much for your time.

WJ