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# 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:

where is the volume, the magnetic field strength, and the magnetic flux density.

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

where is the magnetic permeability of free space, and the relative magnetic permeability.

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Constants

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 at any instant is:

where is the number of turns, and the magnetic flux.

If the current at any instant is , then the instantaneous power () is:

The energy () released from the coil in a time is:

or, by considering (1):

The total energy stored in the coil then becomes:

In order to further define the energy stored in a magnetic field, consider a magnetic circuit of length and cross-sectional area , as diagramed in Figure 1.

We know that the magnetic flux density can be defined as:

from which:

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

which can also be written as:

We know that if the magnetic field strength is uniform, then:

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

where () is the volume. Although this equation was proved for a toroid, it can in fact be demonstrated for all magnetic circuits.

For a curve as the one diagramed in Figure 2, is the blue shaded area:

It can be noted that, if there is no magnetic saturation (i.e. the curve is straight), then:

We also know that the magnetic field strength is related to the magnetic flux density with the equation:

or:

Instant calculator eq(16)

where is the magnetic permeability of free space, and the relative magnetic permeability.

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