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# 10.17: Energy Stored in a Magnetic Field

• • Contributed by Jeremy Tatum
• Emeritus Professor (Physics & Astronomy) at University of Victoria

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.