Calculating Optimal Radius for Cylindrical Vacuum Capacitor

[scotshocker]

Homework Statement

We want to design a cylindrical vacuum capacitor of a given radius a for the outer cylindrical shell, which will be able to store the greatest amount of electrical energy, subject to the constraint that the electric field strength at the surface of the inner sphere may not exceed Eo
(a) What radius b should be chosen for the inner cylindriclal conductor?
(b) How much energy can be stored per unit length

Homework Equations

E=[itex]{λ}/{2πεEo}[/itex]
λ=2πεrEo

The Attempt at a Solution

Electric potential difference:
Vb-Va=∫Edl=∫EdA=∫Edr=[itex]λ/2πε[/itex]∫1/r=([itex]λ/2πε[/itex])*ln([itex]\frac{a}{b}[/itex])
Electrical Energy in a capacitor:
U=[itex]\frac{1}{2}[/itex]λΔV=[itex]\frac{1}{2}[/itex](2πεrEo)*([itex]λ/2πε[/itex])*ln([itex]\frac{a}{b}[/itex])
I would take the derivative of this with respect to b to find the radius. I am not sure that I have set this up correctly.

 

[rude man]

r = b, right? Try to take the derivative wrt b then.

I did not check your algebra.

Also, lambda needs to be substituted for via your 2nd 'relevant equation'. It's a function of b and Eo.