Question

Hello,
I'm using Halliday and Resnick's Fundamentals of Physics, 10th edition. It has some equations in it that seem to contradict each other. I know the problem is with me though, not the equations. Electric potential energy is given as U=VQ (equation 24-3), where Q is charge and V is voltage. We also have the equation, Q=CV (equation 25-1), where C is capacitance. So, it looks like we could say U=CV^2 (substituting CV from 25-1 for Q in 24-3). I know that we cannot, since U=1/2CV^2 (equation 25-22), but I don't see where the problem in my reasoning is.... any help would be appreciated! (Barney style please!)   Thanks!

Darrell

Answer 1

The condition where we apply U=qV

U is the potential energy gained by a particle of charge q, when it is displaced from a position where potential is V_{​​​​​​1} to the position where potential is V_{​​​​​​2} such that V=V_{​​​​​​2}​​​​​ - V_{​​​​1}

In the above condition it is assumed that potential in space is defined, irrespective of the charge q. In other words, the potential exist even if the charge q was not in the picture. And U is the potential energy of the charge q.

The condition where we apply q=CV

Suppose we have two objects separated in space by some distance. These objects initially have equal amount of positive and negative charges, i.e. initially they both have q=0. We move some charge from the one object to the another. A charge q appears on one object, and -q appear on the other. They make some electric field in the space around them, or a potential around them. V is the potential difference from one object to the another. It depends on charge q, as you move more amount of charge, greater electric field magnitude are developed around the objects. In fact, potential V and charge q in this condition are proportional: q=CV.

Note, the difference in the definitions of various quantities involved in the two cases. q in one case is the charge of a point particle which was being moved by a potential difference V. In the other one q was the charge on one object (the charge -q on the other object).

In one case V was already present, it was due to some other charge configuration and not the point particle of charge q itself. In the capacitor case, there was no other charge configuration, V was due to charge +q on one object and -q on the other.

Potential energy U in case of point particle of charge q moved by the potential difference V is U=qV

In the capacitor case, U is defined as the total potential energy of the charge configuration (+q on one object, -q on another).