• https://me.yahoo.com

# Stored Energy

Energy stored in a magnetic field, also considering the case of no magnetic saturation

## Overview

Key facts

The energy stored in a magnetic field is given by:

$E_{stored}&space;=&space;V&space;\int&space;H&space;dB$

where $\inline&space;V$ is the volume, $\inline&space;H$ the magnetic field strength, and $\inline&space;B$ the magnetic flux density.

In the particular case of no magnetic saturation, the energy stored becomes:

$E_{stored}&space;=&space;V&space;\frac{B^2}{2&space;\mu_0&space;\mu_r}$

where $\inline&space;\mu_0$ is the magnetic permeability of free space, and $\inline&space;\mu_r$ the relative magnetic permeability.

<br/>

Constants

$\mu_0&space;=&space;4&space;\pi&space;\cdot&space;10^{-7}&space;\;&space;\frac{N}{A^2}$

If we are to neglect the resistance of the circuit wire, then there would be no energy loss in maintaining a magnetic field. However, energy is required to establish the field, and it can then be recovered when the field is destroyed.

For a toroid, the induced voltage $\inline&space;e$ at any instant is:

$e&space;=&space;N&space;\frac{d\Phi}{dt}$

where $\inline&space;N$ is the number of turns, and $\inline&space;\Phi$ the magnetic flux.

If the current at any instant is $\inline&space;i$, then the instantaneous power ($\inline&space;Watts$) is:

$P&space;=&space;e&space;\cdot&space;i$

The energy ($\inline&space;Joules$) released from the coil in a time $\inline&space;dt$ is:

$E&space;=&space;e&space;\cdot&space;i&space;\cdot&space;dt$

or, by considering (1):

$E&space;=&space;N&space;\cdot&space;i&space;\cdot&space;d\Phi$

The total energy stored in the coil then becomes:

$E_{stored}&space;=&space;\int_0^{\Phi_{Max}}&space;N&space;\cdot&space;i&space;\cdot&space;d\Phi$

In order to further define the energy stored in a magnetic field, consider a magnetic circuit of length $\inline&space;l$ and cross-sectional area $\inline&space;A$, as diagramed in Figure 1.

We know that the magnetic flux density $\inline&space;B$ can be defined as:

$B&space;=&space;\frac{\Phi}{A}$

$\Phi&space;=&space;B&space;\cdot&space;A$

from which:

$d\Phi&space;=&space;A&space;\cdot&space;dB$

Taking into account equations (8) and (5), we obtain the energy stored in the magnetic circuit:

$E_{stored}&space;=&space;\int&space;N&space;\cdot&space;i&space;\cdot&space;A&space;\cdot&space;dB$

which can also be written as:

$E_{stored}&space;=&space;\int&space;\frac{N&space;\cdot&space;i}{l}&space;\cdot&space;A&space;\cdot&space;l&space;\cdot&space;dB$

We know that if the magnetic field strength $\inline&space;H$ is uniform, then:

$H&space;=&space;\frac{N&space;i}{l}$

Taking into account (11), equation (10) becomes:

$E_{stored}&space;=&space;V&space;\int&space;H&space;dB$

where $\inline&space;V$ ($\inline&space;=A&space;l$) is the volume. Although this equation was proved for a toroid, it can in fact be demonstrated for all magnetic circuits.

For a $\inline&space;BH$ curve as the one diagramed in Figure 2, $\inline&space;\int&space;H&space;dB$ is the blue shaded area:

It can be noted that, if there is no magnetic saturation (i.e. the $\inline&space;BH$ curve is straight), then:

$\int&space;H&space;dB&space;=&space;\frac{1}{2}&space;H&space;B$

We also know that the magnetic field strength $\inline&space;H$ is related to the magnetic flux density $\inline&space;B$ with the equation:

$B&space;=&space;\mu_0&space;\mu_r&space;H$

or:

$H&space;=&space;\frac{B}{\mu_0&space;\mu_r}$

where $\inline&space;\mu_0$ is the magnetic permeability of free space, and $\inline&space;\mu_r$ the relative magnetic permeability.

Taking into account equations (15), (13), and (12), the energy stored in this particular case becomes:

$E_{stored}&space;=&space;V&space;\frac{B^2}{2&space;\mu_0&space;\mu_r}$

Last Modified: 3 Aug 10 @ 19:45     Page Rendered: 2022-03-14 15:51:00