# Magnetic field energy

A magnetic field contains energy distributed over the space occupied by the field. It corresponds to the work expended by an electric current when creating a field.

Fig. 2.4

Let the magnetic field be created by the current flowing through the winding of the toroidal coil with the number of turns w and resistance R (Fig. 2.4). When the winding is connected to a power source, the current in it will not be established immediately, but gradually, because magnetic flux covering windings induces self-induction EMF

which prevents the rise of current. The Kirchhoff equation for the electrical circuit of the winding has the form

The energy coming from the power source dWq in time dt is

Part of this energy is dissipated as heat in the resistance of the winding, and another part

if no work is done, it increases the energy of the magnetic field. The magnetic field strength on the axis of the coil in accordance with (1.18) is equal to / H iw l =.

Therefore, the current strength can be represented as. At the same time, the flux linkage of the toroidal coil is S i = H l /w ,Ψ = wΦ = wBs. Substituting expressions for flux linkage and current in (2.1), we find the magnetic field energy in the form

If both parts are divided by V, then we obtain an expression for the energy density, which is also true for an inhomogeneous magnetic field, because any field within infinitesimal limits can be considered as homogeneous, i.e. B

In the event that the magnetic permeability of the medium at all points of the magnetic field is the same (const aμ =), then the energy density is

When the magnetic field  can also be expressed in terms of inductance

where is the final value of the current. This allows us to determine the inductance as a specific measure of the energy of the magnetic field, because it is equal to its double value at a unit current.