Question: We shall charge a capacitor from its initial potential difference V = V, = 0 to a final potential difference V = Vf. Part 1) How much work is done to add the first small amount of charge dq to the capacitor (at V; = 0) and how much work is done to add the last small\" amount of charge dq to the capacitor (at V = Vf)? 1\": (We: (*lncidentally, this question gives us some insight into the use of calculus in the world. Here, dq is obviously not infinitesimal. The minimum charge transfer onto a capacitor is e, which is small, but not infinitesimal.) Part 2) Using the ideas in part1 and the definition of capacitance C, write an integral for the work done to charge the capacitor from uncharged to V = Vf. Your integral should be in terms of V and should not include dq. W = f0\" |:l Part 3) Hence write an expression for the potential energy stored on a capacitor with capacitance C charged to Vf. Part 4) Assume that the capacitor is a parallel plate capacitor with area A, plate separation d, negligible edge effects and no dielectric. Using your answer to part 3), derive an expression for the energy per unit volume of the vacuum between the plates. Give your answer in terms of the electric field E between the plates and do not include C in the expression; simplify it as much as possible. U |:| volume _