# Stored Energy

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

### 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. <br/>

**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:
which leads to:
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: