10.17: Energy Stored in a Magnetic Field

Recall your derivation (Section 10.11) that the inductance of a long solenoid is $\mu n^2 Al$. The energy stored in it, then, is $\frac{1}{2}\mu n^2 AlI^2$. The volume of the solenoid is $Al$, and the magnetic field is $B = \mu n I$, or $H = n I$. Thus we find that the energy stored per unit volume in a magnetic field is

$$\label{10.17.1}\frac{B^2}{2\mu}=\frac{1}{2}BH = \frac{1}{2}\mu H^2.$$

In a vacuum, the energy stored per unit volume in a magnetic field is $\frac{1}{2}\mu_0H^2$- even though the vacuum is absolutely empty!

Equation 10.16.2 is valid in any isotropic medium, including a vacuum. In an anisotropic medium, $\textbf{B}\text{ and }\textbf{H}$ are not in general parallel – unless they are both parallel to a crystallographic axis. More generally, in an anisotropic medium, the energy per unit volume is $\frac{1}{ 2} \textbf{B}\cdot \textbf{H}$.

Verify that the product of $B\text{ and }H$ has the dimensions of energy per unit volume.