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10.17: Energy Stored in a Magnetic Field

Last updated

Mar 5, 2022
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- 10.16: Energy Stored in an Inductance
- 11: Dimensions

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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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