I have forgotten
my Password

Or login with:

  • Facebookhttp://facebook.com/
  • Googlehttps://www.google.com/accounts/o8/id
  • Yahoohttps://me.yahoo.com

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 V is the volume, H the magnetic field strength, and 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 \mu_0 is the magnetic permeability of free space, and \mu_r the relative magnetic permeability.

<br/>

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 e at any instant is:

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

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

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

746/img_em8.png
+
Figure 1

We know that the magnetic flux density 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 H is uniform, then:

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

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

746/img_em16.png
+
Figure 2

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

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

or:

Instant calculator eq(16)
 

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

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