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Yesterday I had a (multiple choice) exam and the following was one of the questions.

We have a coaxial cable (basically two coaxial conducting cylinders) with the inner radius of $a$ (variable) and outer radius of $b$ (constant) filled with vacuum. Find the radius $a$ so that the maximum energy is stored in the cable before it reaches the breakout field $E_b$.

The thing is that, non of the answer choices included the $E_b$ parameter in them. So I thought the question might be wrong. Any answer I can think of must include $E_b$. Am I missing something or the question is wrong?

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Since this question seems to be of the "homework and exercises" variety, I will, in keeping with the policy on this site, initially give just a pointer to the solution, rather than a fully worked example. See how far you get with this.

Regardless of what the breakdown (not breakout) voltage is, there is an optimal solution for radius to maximize energy stored. The correct approach should do the following two things:

1) compute the energy stored at a given voltage, and for a given value of $a$ (assuming fixed value of outer conductor radius $b$). Recall that $E = \frac12 C V^2$ and for a cylindrical capacitor, $C \propto \frac{1}{\log{(b/a)}}$

2) compute the electric field at the surface of the inner conductor (the largest field strength) at a given voltage V

Now you use the formula you found in (2) and substitute it into (1) - you end up with the energy as a function of breakdown voltage and radius. Find the maximum in that expression. While the value of the energy will be a function of the breakdown field, the radius is not.

Update your question with the work if this doesn't solve it (or even if it does...)

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