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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.

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