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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:

E_{stored} = V \int H dB

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

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

E_{stored} = V \frac{B^2}{2 \mu_0 \mu_r}

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

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Constants

\mu_0 = 4 \pi \cdot 10^{-7} \; \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 e at any instant is:

where \inline N is the number of turns, and \inline \Phi the magnetic flux.

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

The energy (\inline Joules) released from the coil in a time \inline dt 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 \inline l and cross-sectional area \inline A, as diagramed in Figure 1.

746/img_em8.png
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Figure 1

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

which leads to:

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 \inline H is uniform, then:

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

where \inline V (\inline =A 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 BH curve as the one diagramed in Figure 2, \inline \int H dB is the blue shaded area:

746/img_em16.png
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Figure 2

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

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

or:

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

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