# Energy product

The energy product results from the magnetic flux density and the magnetic field strength of a magnet and is therefore a variable that serves as a measure of the magnetic energy of a magnet. The individual elementary magnets are all aligned and thus form a magnetic moment. Through this potential energy of all magnetic moments the magnetic energy comes to existence. The greater this energy, the larger the energy product and the greater the forces of the magnet.

The so-called hysteresis curve visualizes the relationship between the magnetic flux density and the magnetic field strength during demagnetization or magnetization. There are several special features of this curve: For example, the remanence flux density or the remanence can be identified very well. The term remanence is understood to be the magnetization of the material present after removal of an external magnetic field. If an object magnetized in this way is to be demagnetized again by means of a magnetic field, the so-called coercive field is necessary for this purpose. This field is a magnetization-opposing magnetic field with a certain coercive field strength. From this strength the magnetization is canceled, but not vice versa. The energy product is abbreviated by the symbol E and can be calculated from the maximum product of the magnetic field strength H and the magnetic flux density B. Thus:

The energy product can also be determined from the magnetic field strength to the product with the flux density. However, the resulting result is about four times larger than the actual maximum energy product. Furthermore, a proportional relationship between the energy density w (ie the amount of energy per unit volume of the magnet) and the energy product applies. If the energy density is calculated exactly, then it turns out that the proportional relation to the energy product is just the factor 0.5:

Mathematically correct, however, the energy density w would increase over the integral the magnetic field strength H above the flux density B are determined:

Although the relationship described in (2) is not exact, it approximately fulfills the requirements of a magnet whose magnetic field strength is proportional to the magnetic flux. It also applies here that the local derivation of the energy product is proportional to the force: this is imaginable due to the force density acting along one direction. This force density is at the same time the energy density change in the same direction.

If the energy density, ie the energy per unit volume, is multiplied by the volume of the magnet, the total magnetic energy W stored in the magnet is obtained. On the other hand, with this volume, of course, half of the energy product can be multiplied - the result is the same:

From the formulas also follows that the unit for the energy product is the product of Oersted (A / m) and Tesla (N / Am). The abbreviated results in the unit J / m³, or N / m²: The energy product is therefore used to calculate the force between two ferromagnetic materials abutting or repelling at a known pole face. For the pole face A, the energy product E and the magnetic force F:

From this formula, some dependencies can be deduced: For example, the force between the two magnets doubles when either the energy product or the pole surface is doubled. The magnetic flux density in a permanent magnet is equal to the remanence or the so-called B-field. With the remanence the magnetization of the material is indicated, as already briefly mentioned above. Here, the magnetic field H of the magnet is in a proportional relationship to the remanence - of course, material-specific properties must be considered. These are influenced by the factors μ (magnetic permeability of matter) and (magnetic permeability in vacuum):

If H is now inserted in (1):

Thus, the energy density of the magnet is proportional to the square of the remanence - so if the magnetization is twice as strong, four times as much magnetic energy is stored in the material. Conversely, this means that double magnetization can increase magnetic forces fourfold.

**Physical explanation**

The elementary magnets, which clearly serve to clarify magnetization processes in physics education, are basically the electron spins of the free electrons of each atom in the ferromagnetic material. If the atomic electron spins are also aligned twice as strongly with a doubling field for the magnetization, then they are also attracted twice as much. Thus, the total amount of energy of the magnet is four times larger for a field twice as strong.

Every system generally tries to reach an energetic minimum. Earlier, the location derivation of the energy was mentioned: If we were outside an energetic minimum, the location derivation always points to the place where the energy minimum is located. However, if we are directly at this minimum, the derivative is undefined and disappears. According to this understanding, magnetic forces work out of the endeavor of a system in ferromagnetic materials to strive for the lowest possible energy level. Another insight can be gained by using (7) in (5):

It follows that the force between two magnets is proportional to the square magnetic flux and the cross-sectional area. At a large μ, the energy density due to the fracture becomes particularly small. Ferromagnetic materials usually have a very large μ (between 1000 and 10,000 for iron for example). As the magnet moves away from the iron, the energy density of the surrounding air increases. It becomes larger than the energy density that would be present if the field lines were to run directly through the iron. In order to find the balance again, the system strives for the energetic minimum: According to this, as many field lines as possible should be in the iron. This quest for energetic balance is expressed in the force that moves the magnet back to the iron.