Energy In A Magnetic Field

Consider an inductor connected to a variable source of EMF. The inductor opposes any change in the current passing through it; hence, an EMF is induced across it.

Now, consider an ideal coil with no resistance; no energy is dissipated across it. Hence, some amount of the energy supplied by the source is stored in the inductor.

The instantaneous power is given by the product of the instantaneous induced EMF and the instantaneous current. The energy supplied to the inductor is its product with the differential time. Assuming that the energy stored is zero when there is no current, integrating the expression gives the energy stored in the magnetic field.

In an ideal toroidal solenoid, assuming a small cross-sectional area through which the magnetic field is constant, the inductance is known and so is its volume. Thus, the magnetic energy density can be calculated.

If the material inside the toroid is not a vacuum but given by the magnetic permeability μ, the expression is modified thus.