## 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?

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

1. Ned Wright Says:

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. Phillip Helbig Says:

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”. 🙂