# Energy of a magnetic field

The **energy of the magnetic field** results from the excitation of the space permeated by the magnetic field. It can be thought of as the potential energy that would be imparted on a charged particle moving through a region with an external magnetic field present.

## Energy stored in an inductor

As a result of the induced magnetic field inside an inductor of inductance $L$ when a current, $i,$ flows through, energy is said to be stored in the magnetic field of the inductor.

$U=\frac12Li^2$

An LC oscillator consists of an inductor and a capacitor passing energy back and forth without dissipation. A capacitor with capacitance $C$ is initially charged to $q_0$ before being attached to an inductor of inductance $L.$ What is the current when the charge has decreased to half its initial value?

Since energy is not dissipated, employ conservation of energy.

$U_{L_i} + U_{C_i} = U_{L_f} + U_{C_f}$ $0 + \frac{q_0^2}{2C} = \frac12Li^2 + \frac{(0.5q_0)^2}{2C}$

$\rightarrow i^2 = \frac{3q_0^2}{4LC}$

Since the natural frequency of an LC oscillator is $\omega = \frac{1}{\sqrt{LC} },$

$i = \frac{\sqrt{3}}{2} q_0 \omega$

## Energy of a magnetic field

The magnetic field component of an electromagnetic wave carries a magnetic energy density $u_B$ given by

$u_B = \frac{B^2}{2\mu_0}$

where $B$ is the amplitude of the magnetic field and $\mu_0=4\pi \times 10^{-7} \frac{\text{N}}{\text{m}^2}$ is the permeability of free space.

What is the magnetic energy density of an electromagnetic wave with electric field amplitude $E = 300,000 \frac{\text{N}}{\text{C}}?$

First, express the magnetic field amplitude in terms of the electric field amplitude.

$c=\frac{E}{B}$

$\Rightarrow B = \frac{E}{c}$

Then, calculate the energy density.

$\begin{aligned} u_B &= \frac{B^2}{2\mu_0} \\ &= \frac{E^2}{2\mu_0 c^2} \\ &=\frac{(3\times 10^5)^2}{2(4\pi \times 10^{-7})(3\times 10^8)^2} \\ &=\frac{5}{4\pi} \frac{\text{J}}{\text{m}^3} \text{ }_\square \\ \end{aligned}$

**Cite as:**Energy of a magnetic field.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/energy-of-a-magnetic-field/