Energy stored in a magnetic field, also considering the case of no magnetic saturation
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Gyroscopic Couple: The rate of change of angular momentum () = (In the limit).
= Moment of Inertia.
= Angular velocity
= Angular velocity of precession.
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.
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:
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:
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: