Energy in the magnetic field

To calculate the energy stored in a coil, the toroidal coil illustrated in Figure 1 is connected to a DC voltage source. At time \(t = 0\), the voltage source is switched on. The voltage across the coil is therefore equal to the source voltage \(U\) at every point in time \(t > 0\). The current flowing through the coil will increase linearly according to equation ??. The energy supplied in this way generates a flux density in the coil core. To calculate the energy supplied to the coil, the known equation for electrical power ?? can be transformed and then the voltage can be replaced by equation ??. As a result of this transformation, the change in magnetic energy \(\mathrm {d}W_{\mathrm {L}}\) is obtained, which is calculated from the multiplication of inductance \(L\), the current through the inductance \(i_{\mathrm {L}}\) and the rate of change of the current through the inductance \(\mathrm {d}i_{\mathrm {L}}\) (see equation ??).

\begin {align} p &= \frac {\mathrm {d}W}{\mathrm {d}t} = u\cdot i \label {GlArbeit}\\ u &= L\cdot \frac {\mathrm {d}i}{\mathrm {d}t} \tag {\ref {GLInduktivitaet2}}\\ \mathrm {d}W_{\mathrm {m}}&=u_{\mathrm {L}}\cdot i_{\mathrm {L}} \cdot \mathrm {d}t = L\cdot \frac {\mathrm {d}i}{\mathrm {d}t}\cdot i_{\mathrm {L}}\cdot \mathrm {d}t = L\cdot i_{\mathrm {L}} \cdot \mathrm {d}i_{\mathrm {L}} \label {GlÄnderungmagnetischeEnergie} \end {align}

The total energy is the integral over \(\mathrm {d}W_{\mathrm {m}}\) from the initial value \(i_{\mathrm {L}}=0\) to the static final value \(i_{\mathrm {L}} = I\). Assuming a constant inductance \(L\), the following applies:

\begin {equation} W_m = L \cdot \int _0^{I_L} i_L \cdot \mathrm {d}i_L = \frac {1}{2} \cdot L \cdot I^2 = \frac {1}{2} \cdot N \cdot \varPhi \cdot I \end {equation}

The magnetic energy can also be calculated from the field magnitudes by integration with the magnetic flux density: \begin {equation} W_{\mathrm {m}} = \ell _{\mathrm {m}}\cdot A\cdot \int _0^{B_L} H\mathrm {d}B = \ell _{\mathrm {m}}\cdot A \cdot \frac {B_{\mathrm {L}}^2}{2\cdot \mu _{\mathrm {r}}\cdot \mu _0} \end {equation} In an electromagnet, the energy density in the air gap (i.e. the energy per volume of the air gap) is particularly important: it represents the „force of the electromagnet“. The maximum force is exerted precisely at the transition from the air gap to the iron core – corresponding to a theoretically infinitely narrow air gap. Since the air gap, as the name suggests, is usually filled with air, \(\mu _{\mathrm {r}} = 1\) in normal cases and can be omitted.

\begin {equation} \frac {W_{\mathrm {m}}}{V} = \frac {B_{\mathrm {L}}^2}{2\cdot \mu _0} = \frac {F}{A} \end {equation}