Magnetic and electric energies in an LC circuit (21-27)

Review

We recognize the second term on the right-hand side of Equation 21-27,$q^2/2C$, from Section 17-6: It’s just the electric potential energy stored in the capacitor. Since a capacitor plays the same role in an $LC$ circuit as the spring does in a block–spring combination, this electric potential energy is analogous to the elastic potential energy of a spring. We use the symbol $U_E$ for this electric energy. The subscript $E$ reminds us that an $\vec{E}$-field is involved; you can think of $U_E$as the energy required to take an uncharged capacitor and move charges $+q$ and $-q$ to the plates of the capacitor, thereby setting up an electric field $\vec{E}$ between the plates.

The first term on the right-hand side of Equation 21-27, $(1/2)Li^2$, is one that we haven’t seen before. The presence of the inductance $L$ in this term tells us that it represents energy stored in the inductor as a result of the presence of current. To see how this energy arises, recall from Section 21-4 that an inductor sets up an emf that opposes any change in the current through the inductor. If we want to make current flow through an inductor where none was flowing before, we have to do work against that emf. The quantity $(1/2)Li^2$ is exactly equal to the amount of work we have to do. By building up the current from zero to $i$, we also create a magnetic field $\vec{B}$, so we can think of $(1/2)Li^2$ as the energy required to set up a magnetic field in and around the inductor. That’s why we call this magnetic energy and denote it by the symbol $U_B$ (the subscrip $B$ reminds us that a $\vec{B}$-field is involved).