## A Problem of Capacitors

Time for another entry in the Cute Problems category. I’ve been teaching a ~~course~~ module in theoretical physics this term so here’s one that my students should find a doddle…

*A spherical capacitor consists of an outer conducting sphere of fixed radius b and a concentric inner conducting sphere whose radius a can be varied. The space between the spheres is filled with air which has a breakdown electric field strength E _{0}. What are the greatest achievable values for (i) the potential difference between the spheres, and (ii) the electrostatic energy stored in the capacitor?*

Answers via the comments box please.

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April 4, 2014 at 4:20 am

cgs units for Q:

max voltage with inner radius at breakdown is

E = E_0*(a^2/r^2) so V = E_0*a(1-a/b) maximized for

1-2a/b=0 or a=b/2. Hence V_max=0.5*E_0*a = 0.25*E_0*b

max energy is at a larger inner radius to get more charge:

Energy = 0.5*V*Q, and Q = a^2*E_0, so

U= 0.5*a^2*a*(1-a/b)*E_0^2

= 0.5*a^3*(1-a/b)*E_0^2

maximized when 3*a^2-4*a^3/b=0 so

a/b = 3/4 and

U = 0.5*(27/64)*(1/4)*b^3*E_0^2

= (27/512)*b^3*E_0^2

April 4, 2014 at 10:07 am

Your blog is a good source for new collective nouns. A while back we had “a compression of distances”. Now we have “a problem of capacitors”. 🙂