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# Stored Energy

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

Gyroscopic Couple: The rate of change of angular momentum () = (In the limit).
• = Moment of Inertia.
• = Angular velocity
• = Angular velocity of precession.

## 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 (2):

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 (9) and (6), 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 (12), equation (11) 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:

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

Taking into account equations (16), (14), and (13), the energy stored in this particular case becomes: