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Energy product

What is the energy product?

The energy product is a measure of the magnetic energy stored in a magnet. This magnetic energy is created by the potential energy of all aligned magnetic moments. The greater the energy product, the greater the magnetic forces emanating from the magnet. It is called energy "product" because it is determined as the product of magnetic field strength and magnetic flux density.
Table of Contents
The energy product E of a magnet is the maximum product of magnetic flux density B and magnetic field strength H which can be present in the material at the same time. Therefore, the following applies: E=B•H.

The relationship between magnetic field strength and magnetic flux density during magnetisation or demagnetisation is described by the so-called hysteresis curve. If you look at the hysteresis curve, you will notice the remanence or remanence flux density, for example. This is the remaining magnetisation in the material when no external fields are present. The magnetic field strength required to make the magnetic flux density in the material disappear is the so-called coercive field strength.

You can find these specifications as valid values in the overview table of physical magnet data.

Calculation of the energy product

To calculate the energy product, you cannot simply multiply the magnetic remanence flux density by the coercive field strength listed in this table. If you multiply these two values, the result is about a factor of four greater than the "true" maximum energy product. The following diagram of typical hysteresis curves is the best way to illustrate the stated variables and the calculation of the maximum energy product:

Hysteresis curves for a magnetically soft material (left) and a magnetically hard material (right). For the still non-magnetised material, the red
Hysteresis curves for a magnetically soft material (left) and a magnetically hard material (right). For the still non-magnetised material, the red "initial magnetisation curve" shows the course of the magnetisation over the external field. The upper curve shows the progression from the saturation flux density BS to – BS, i.e. demagnetisation, and the lower curve shows the progression from – BS to BS, i.e. magnetisation, as indicated by the arrows. Typical points on the hysteresis curve are the coercive field Hc, which is necessary to compensate for the magnetisation of the material by the external field, the remanence BR, which denotes the remaining flux density when the external field disappears, and the saturation flux density BS, at which all electron spins are aligned. While the product of BR and Hc is shown in the first quadrant (between 0 and 3 o'clock) of both graphs (as a black rectangle), the maximum energy product is determined via demagnetisation. You have a material and measure the energy it contains through a demagnetisation process. By definition, the maximum energy product is the largest possible rectangle (product) of H and B that "fits" under the hysteresis curve in the 4th quadrant (and symmetrically in the 2nd quadrant). As you can see, the area of this maximum energy product is significantly smaller than the product of BR and Hc

The energy product is proportional to the amount of energy stored per unit volume in a magnet. This amount of energy per volume of the magnet is the energy density w. The exact calculation of the energy density shows that in the simplest case of magnetisation, which increases proportionally to the magnetic field, it is just half of the energy product:

w=\frac{1}{2} \cdot {E}= \frac{1}{2} \cdot {B} \cdot {H}
The total amount of magnetic energyW in a magnet is the product of energy density w and volume V (W=w•V). If you multiply half of the energy product by the volume of the magnet, you arrive at the total amount of energy stored in a magnet:

W=w \cdot {V} = \frac{1}{2} \cdot {E} \cdot {V}= \frac{1}{2} \cdot {B} \cdot {H} \cdot {V}
The amount of magnetic energy in a permanent magnet depends on the product of the magnetic flux density B and the magnetic field H, as well as the volume of the magnet V. The magnetic energy is the potential energy of all aligned elementary magnets in the material that generate the magnetic flux.
The amount of magnetic energy in a permanent magnet depends on the product of the magnetic flux density B and the magnetic field H, as well as the volume of the magnet V. The magnetic energy is the potential energy of all aligned elementary magnets in the material that generate the magnetic flux.
The unit for the energy product is the product of Tesla (N/Am) and Oersted (1 Oe = 79,577 A/m). This results in a unit of the dimension N/m² or J/m³, i.e. the dimension of energy per volume.

From the energy product of a magnet and the area of the north or south pole, the force between two magnets or the force between a magnet and a ferromagnetic material (e.g. iron) can be loosely calculated. For two cylindrical magnets with the pole area A and the energy product E, the following applies to the magnetic force F:

F = A • E

This means that if you double the adhesion area of a magnet at the same amount of energy per volume (described by the energy product), the force with which the magnet adheres to an iron plate doubles. However, if you double the energy product at the same volume and the same adhesion surface, the force is also doubled.
In a permanent magnet, the B field, i.e. the magnetic flux density, is equal to the remanence. The remanence indicates the magnetisation present in the material. The magnetic field H in the permanent magnet is proportional to the remanence but takes material properties such as the magnetic permeability μ into account. The following applies:

\(H=\frac{1}{\mu\cdot\mu_0} \cdot{B}\)
Thus, the following applies:

\(E= B \cdot {H} =\frac{1}{\mu\cdot\mu_0} \cdot{B^2}\)
The energy density of a magnetised material is therefore proportional to the square of the remanence. With double magnetisation, four times the amount of magnetic energy is stored in the material. This means that with double magnetisation, the forces of a magnet increase fourfold. You can think of it this way:
If you double the magnetic field of a magnet, the atomic spins are aligned "twice" as strongly when a material is magnetised in the field of this magnet. Each of these spins acts like an elementary magnet and is, in turn, attracted twice as strongly. This means that the total force effect and the total amount of energy in the magnet are four times greater when the field is doubled.
Mathematically, the energy density w is determined as the integral of the magnetic field strength H over the magnetic flux density B:

\(w=\int{HdB}\)
The relationship w = 1 / 2 • B • H between B-field, H-field and the energy density w is only obtained for magnets whose magnetic flux B is proportional to the magnetic field strength H. Although this is usually not exactly the case, it is often approximately true.

The force density along a direction is the change in energy density along this direction. The force is, therefore, proportional to the spatial derivative of the energy product.

This idea corresponds exactly to the picture that every system generally strives towards an energetic minimum. Outside of an energetic minimum, the spatial derivative of the energy points to the spot where the minimum energy is located. At the location of the minimum, however, the derivative disappears. The cause of the action of magnetic forces can also be understood as the effort of a system of magnets and ferromagnetic materials to strive for an energetic minimum.
If you insert the magnetic flux density B and the magnetic permeability μ into the force formula F = A • E, the following applies to the magnetic force F:

\(F=\frac{1}{2\cdot\mu\cdot\mu_0} \cdot {A} \cdot{B^2}\)
The force is, therefore, proportional to the cross-sectional area A and proportional to the square of the magnetic flux density B of a magnet.

Because of \(w=\frac{1}{2\cdot\mu\cdot\mu_0} \cdot{B^2}\), the energy density is particularly low when μ is high. For ferromagnetic materials, μ is very high (e.g. 1 000 – 10 000 for iron). If the magnet is moved away from the iron, the energy density of the air surrounding the magnet is greater than the energy density would be if the field lines of the magnet were to pass through iron. The system is thus not at the energy minimum, providing that as many field lines as possible do not pass through the iron. This manifests itself as a force trying to move the magnet back towards the iron.


Portrait of Dr Franz-Josef Schmitt
Author:
Dr Franz-Josef Schmitt

Dr Franz-Josef Schmitt is a physicist and academic director of the advanced practicum in physics at Martin Luther University Halle-Wittenberg. He worked at the Technical University from 2011-2019, heading various teaching projects and the chemistry project laboratory. His research focus is time-resolved fluorescence spectroscopy in biologically active macromolecules. He is also the Managing Director of Sensoik Technologies GmbH.

The copyright for all content in this compendium (text, photos, illustrations, etc.) remains with the author, Franz-Josef Schmitt. The exclusive rights of use for this work remain with Webcraft GmbH, Switzerland (as the operator of supermagnete.gr). Without the explicit permission of Webcraft GmbH, the contents of this compendium may neither be copied nor used for any other purpose. Suggestions to improve or praise for the quality of the work should be sent via e-mail to [email protected]
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