Stored Energy

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Key facts

The energy stored in a magnetic field is given by:

$$E_{stored} = V \int H dB$$
(1)

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:

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

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

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Constants

$$\mu_0 = 4 \pi \cdot 10^{-7} \; \frac{N}{A^2}$$
(3)

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

Physics Magnetism

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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:

$$e = N \frac{d\Phi}{dt}$$
(4)

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:

$$P = e \cdot i$$
(5)

The energy ($Joules$) released from the coil in a time $dt$ is:

$$E = e \cdot i \cdot dt$$
(6)

or, by considering (4):

$$E = N \cdot i \cdot d\Phi$$
(7)

The total energy stored in the coil then becomes:

$$E_{stored} = \int_0^{\Phi_{Max}} N \cdot i \cdot d\Phi$$
(8)

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.

Figure 1
Figure 1

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

$$B = \frac{\Phi}{A}$$
(9)

which leads to:

$$\Phi = B \cdot A$$
(10)

from which:

$$d\Phi = A \cdot dB$$
(11)

Taking into account equations (11) and (8), we obtain the energy stored in the magnetic circuit:

$$E_{stored} = \int N \cdot i \cdot A \cdot dB$$
(12)

which can also be written as:

$$E_{stored} = \int \frac{N \cdot i}{l} \cdot A \cdot l \cdot dB$$
(13)

We know that if the magnetic field strength $H$ is uniform, then:

$$H = \frac{N i}{l}$$
(14)

Taking into account (14), equation (13) becomes:

$$E_{stored} = V \int H dB$$
(15)

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:

Figure 2
Figure 2

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

$$\int H dB = \frac{1}{2} H B$$
(16)

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

$$B = \mu_0 \mu_r H$$
(17)

or:

$$H = \frac{B}{\mu_0 \mu_r}$$
(18)

\calc{V*(B^2/(8*π*10^(-7)*mu_r))} "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 (18), (16), and (15), the energy stored in this particular case becomes:

$$E_{stored} = V \frac{B^2}{2 \mu_0 \mu_r}$$
(19)