## 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 E0. 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.

### 2 Responses to “A Problem of Capacitors”

1. 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

2. 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”. 🙂