\(\def \u#1{\,\mathrm{#1}}\) \(\def \us#1{\,\mathrm{\scriptsize #1}}\) \(\def \abs#1{\left|#1\right|}\) \(\def \ast{*}\) \(\def \deg{^{\circ}}\) \(\def \tau{\uptau}\) \(\def \ten#1{\times 10^{#1}}\) \(\def \redcancel#1{{\color{red}\cancel{#1}}}\) \(\def \BLUE#1{{\color{blue} #1}}\) \(\def \RED#1{{\color{red} #1}}\) \(\def \PURPLE#1{{\color{purple} #1}}\) \(\def \th#1,#2{#1,\!#2}\) \(\def \lshift#1#2{\underset{\Leftarrow\atop{#2}}#1}}\) \(\def \rshift#1#2{\underset{\Rightarrow\atop{#2}}#1}}\) \(\def \dotspot{{\color{lightgray}{\circ}}}\) \(\def \ccw{\circlearrowleft}\) \(\def \cw{\circlearrowright}\)
Chapter 15: Electric Potential
9. (Incomplete)

Energy of a Capacitor

Like charges repel each other, so once a conductor has a positive charge, adding more positive charge onto the conductor requires work. If the charge on the conductor is $Q$, then the conductor is a potential of $\Delta V=Q/C$ above background potential. To move an additional little charge $q$ from far away onto the conductor, you need to do work $W=q\Delta V = q{Q\over C}$ on that charge. The total work required to put charge $Q$ on a conductor with capacitance $C$ is $$W=\frac12(Q)\left({Q\over C}\right)$$

This work isn't wasted, mind you: this work stores potential energy inside the conductor, which is released when the charges are allowed to leave. Thus the potential energy of a conductor with capacitance $C$ and total charge $Q$ on it is

$$PE_C={Q^2\over 2C}$$

You can think of this as the potential energy of all those like charges who want to escape from each other. The more charge on the conductor, the larger this potential energy. If the conductor has a larger capacitance (for instance, if it is a larger piece of metal), however, then the potential energy is smaller as the charges have more elbow room.

We can use the relationship $Q=C\Delta V$ to write this potential energy in terms of the potential of the conductor: $PE_C = {(C\Delta V)^2\over 2C}$, or

$$PE_C = \frac12C(\Delta V)^2$$