Revisiting today's class: what we learned about Q and E-fields (and energy)

In today's class, I think we may have learned something interesting  when we began to do problem 40. As you may recall in that problem we were given the energy density in the capacitor, and asked to find the charge, Q.  There was confusion, at first, when we realized that none of the equation that we had just summarized for the capacitor would  help us.  Those equations related V and Q and U and C,  but they did not involve E.  The equation that involved E, which related E to Q, was where we originally began our examination of the capacitor.  (From that we got V and all our other equations.)  For this problem we had to go back to that more fundamental  relation and equation.  Then we were able to solve it using the direct relation between E and Q,  and the relationship between energy density and E.

This illustrates a fairly fundamental and universal  aspect  of electrostatic problems. Very often one naturally starts with the charge distribution, one gets E from that, and then obtains V from E. A natural way to derive things is to go from q (or Q) to E to V.

Realizing that tells us how to do problem 80 (the " in impossible one").  First, you would need to figure out the electric field as a function of r for a given charge, Q;  then you can integrate that, from the inner radius to the outer radius, to obtain  the voltage difference between the inside and outside  conductors.  Capacitance is defined by the relationship, usually linear, between Q  and V.