Atomic currents, magnetic dipoles, and magnetization
In a simple model, an orbiting electron has a magnetic moment proportional to its orbital angular momentum,
m = – e me
2 L ...(1)
and a singular contribution due to the spin angular momentum, m =
e
me S ...(2)
The magnetization in a material is the magnetic moment per unit volume:
M = < ∑m >
V
i
∆ ...(3)
Diamagnetism
In diamagnetism materials, magnetic dipole moments are induced in molecules by the magnetic field, and the vectors M and B have opposite directions.
Paramagnetism
The permanent magnetic moment of an unpaired electron in a paramagnetic substance tends to become aligned with the magnetic field. The vectors M and B are parallel and are related by Curie’s law
M = CB
µ0T ...(4) valid except at low temperatures and high fields.
Ferromagnetism
Molecular magnetic dipoles in a magnetic domain tend to be aligned in a ferromagnetic material. If the domains are oriented preferentially by applying a magnetic field, the sample has a large magnetization. The magnetization can persist in hard magnetic materials to form a permanent magnet.
Magnetic intensity H
The magnetic intensity H is defined by the relation
B = µ0 (H + M) ...(5)
For a linear medium with permeability µ, the relation can be expressed as B = µH. For a Rowland ring, H is due to the macroscopic current in the windings. The relation between B and H for ferromagnetic materials is nonlinear, and hysteresis effects are present.
The magnetic field of the earth
Outside its surface, the earth’s magnetic field is approximately a dipole field. Large changes in the field occur over geological time intervals.
Magnetic field in magnetic materials – Hysteresis
The field of a long solenoid is directly proportional to the current. Indeed the field B0 inside a solenoid is given by
B0 = µ0nI
This is valid if there is only air inside the coil. If we put a piece of iron or other ferromagnetic material inside the solenoid, the field will be greatly increased, often by hundreds or thousands of times. This occurs because the domains in the iron become preferentially aligned by the external field. The resulting magnetic field is the sum of that due to the current and that due to the iron. It is sometimes convenient to write the total field in this case as sum of two terms :
B = B0 + BM ...(6)
Here, B0 refers to the field due only to the current in the wire (the “external field” ); it is equal to the field that would be present in the absence of a ferromagnetic material. Then BM represents the additional field due to the ferromagnetic material itself; often BM >> B0.
The total field inside a solenoid in such a case can also be written by replacing the constant µ0 by another constant, µ, characteristic of the material inside the coil:
B = µnI ...(7)
µ is called the magnetic permeability of the material. For ferromagnetic materials µ is much greater than µ0. Φορ all other materials, its value is very close to µ0 #. The value of µ, however, is not constant for ferromagnetic materials; it depends on the value of the external field B0, as the following experiment shows.
Measurements on magnetic materials are generally done using a torus, which is essentially a long solenoid bent into the shape of a circle (Figure –1), so that practically all the lines of B remain within the torus.
Suppose the torus has an iron core that is initially unmagnetized and there is no current in the windings of the torus. Then the current I is slowly increased, and B0 increases linearly with I. The total field B also increases, but follows the curved line shown in the graph of Figure –2. (Note the different scales: B >> B0) Initially (point a), no domains are aligned. As B0 increases, the domains become more and more aligned until at point b, nearly all are aligned. The iron is said to be approaching saturation. (Point b is typically 70 percent of full saturation; the curve continues to rise very slowly, and reaches 98 percent saturation only when B0 is increased by about a thousand fold above that at point b; the last few domains are very difficult to align). Now suppose the external field B0 is reduced by decreasing the current in the coils. As the current is reduced to zero, point c in Figure – 3, the domains do not become completely unaligned. Some permanent magnetism remains. If the current is then reversed in direction, enough domains can be turned around so B = 0 (point d). As the reverse current is increased further, the iron approaches saturation in the opposite direction (point e). Finally, if the current is again reduced to zero and then increased in the original direction, the total field follows the path efgb, again approaching saturation at point b.
R
Notice that the field did not pass through the origin (point a) in this cycle. The fact that the curves do not retrace themselves on the same path is called hysteresis. The curve bcdefgb is called a hyteresis loop. In such a cycle, much energy is transformed to thermal energy (friction) due to realigning of the domains; it can be shown that the energy dissipated in this way is proportional to the area of the hysteresis loop.
At point c and f, the iron core is magnetized even though there is no current in the coils. These points correspond to a permanent magnet. For a permanent magnet, it is desired that ac and af be as large as possible. Materials for which this is true are said to have high retentivity, and may be referred to as
“hard”. On the other hand, a hysteresis curve such as that in Figure –4 occurs for so-called “soft iron” (it is soft only from a magnetic point of view).
This is preferred for electromagnets since the field can be more readily switched off, and the field can be reversed with less loss of energy. Whether iron is “soft” or “hard” depends on how it is alloyed, heat treatment, and other factors.
A ferromagnetic material can be demagnetized – that is, made unmagnetized.
This can be done by reversing the magnetizing current repeatedly while decreasing its magnitude. This results in the curve of Figure – 5. The heads of a tape recorder are demagnetized in this way; the alternating magnetic field acting at the heads due to a demagnetizer is strong when the demagnetizer is placed near the heads and decreases as it is moved slowly away.
# All materials are slightly magnetic. Non-ferromagnetic materials fall into two classes: paramagnetic, in which µ is very slightly larger than µ0; and diamagnetic, in which µ is very slightly less than µ0. Paramagnetic materials apparently contain atoms that have a net magnetic dipole moment due to orbiting electrons, and these become slightly aligned with an external field just as the galvanometer coil experiences a torque that tends to align it. Atoms of diamagnetic materials have no net dipole moment. However, in the presence of an external field, electrons revolving in one direction are caused to increase in speed slightly, whereas those revolving in the opposite direction are reduced in speed; the result is a slight net magnetic effect which actually opposes the external field.
B0 B
Figure – 4 Hysteresis Curve for soft iron
B0 B
Figure – 5 Successive hysteresis loops
during demagnetization