Ferromagnetism, Physics tutorial

Introduction to Ferromagnetism:

Ferromagnetic materials are such materials that respond much strongly to the presence of magnetic fields. In these materials, the magnetic dipole moment of the atoms occurs due to the spins of unpaired electrons. Such tend to line up parallel to one other. This kind of a line-up doesn't take place over the whole material; however it takes place over a small volume termed as domain. Though, such volumes are big as compared to the atomic or molecular dimensions.

These line-ups occur even in the absence of the external magnetic field. You should be wondering about the nature of forces which cause the spin magnetic moments of various atoms to line up parallel to one other.

1937_alignment direction of the magnetic moment.jpg

In an unmagnetised ferromagnetic material, the magnetic moments of various domains are arbitrarily oriented and the resultant magnetic moment of the material, as an entire is zero, as represented in above figure. Though, in the presence of an external magnetic field, the magnetic moments of the domains line-up in such a way as to provide a total magnetic moment to the material in the direction of the field. The method through which this occurs is that the domains having the magnetic moments in favored directions rise in size at the expense of the other domains. Moreover, the magnetic moments of the total domains can rotate. The material is therefore magnetized. If, subsequent to this, the external magnetic field is decreased to zero, there still remains a considerable quantity of magnetization in the material. The material gets permanently magnetized. The behavior of ferromagnetic materials, beneath the action of changing magnetic fields, is quite complex and represents or shows the phenomenon of hysteresis that literally signifies 'lagging behind'.

Beyond certain temperature, termed as the 'Curie Temperature', as the forces of thermal agitation dominate the exchange forces, the domains lose their dipole moments. The ferromagnetic material starts to behave similar to a paramagnetic material. If cooled, it recovers the ferromagnetic properties.

Ultimately, we are familiar with the two other kinds of magnetism, which are closely linked to ferromagnetism. These are anti-ferromagnetism and ferrimagnetisms (as well termed as ferrites).

In antiferromagnetism substances, the exchange forces play the role of setting up the adjacent atoms into anti-parallel alignment of their equivalent magnetic moments, that is, adjacent magnetic moments are set up in the opposite directions. These substances show little or no evidence of magnetism present in the body. Though, when these substances are heated above the temperature termed as Neel temperature, then the exchange force ceases to act and the substance behaves similar to any other paramagnetic material.

In ferrimagnetic substances, usually termed as ferrites, the exchange coupling locks the magnetic moments of the atoms in the material into a pattern, as shown in the figure below. The external effects of such an alignment are intermediate between ferromagnetism and antiferromagnetism. Here, in this the exchange coupling vanishes above a certain temperature.

2314_ferrimagnetic substances.jpg

Therefore, we discover that the magnetization of the materials is due to the permanent (and induced) magnetic dipoles in such materials. The magnetic dipole moments in such materials are due to the circulating electric currents, termed as amperian currents at atomic and molecular levels.

Magnetic Field due to a Magnetized Material:

We are familiar with the macroscopic properties of dielectric materials in terms of the polarization vector 'P', the origin of which is in the dipole moments of its natural or induced electric dipoles. We take on a parallel procedure in the study of magnetic materials. We can substitute the electric field vector 'E' by 'B', then substitute 'P' by an analogous quantity which we state magnetization vector 'M' which is the magnetic dipole moment per unit volume. Moreover, we substitute the polarization charge density ρp   by magnetic charge density ρm, by writing ∇.M = - ρm just as we had ∇.P = ρp   

We are familiar that the magnetization of matter is due to the circulating currents in the atoms of the materials. This was initially recommended by Ampere, and we state such circulating currents as 'amperian' current loops. Such currents occur due to either the orbital motion of electrons in the atoms or their spins. Such currents, apparently, don't involve big scale charge transport in the magnetic materials as in case of conduction currents. The currents are as well termed as magnetization currents and we associate such currents to the magnetization vector 'M'.

Let take a slab of uniformly magnetized material. It includes a large number of atomic magnetic dipoles (that is, evenly distributed all through its volume) all pointing in the similar direction. If 'µ' is the magnetic moment of each and every dipole, then the magnetization 'M' will be the product of 'µ' and the number of oriented dipoles per unit volume. As dipoles can be pointed by small current loops, so assume that the slab comprises of numerous tiny loops. Suppose any small loop of area 'a'. In terms of magnetization 'M', the magnitude of dipole moment 'µ' is written as:

µ = Madz

Here, dz is the thickness of the slab

If the tiny loop consists of a circulating current 'I', then the dipole moment of the tiny loop is represented by:

µ = Ia

Equating both the above equation, we get:

M = I/dz or I = Mdz

Let us take another case when there is non-uniform magnetization in the material, then the atomic currents in the (amperian) circulating current loops don't have the similar magnitude at all points within the material and, obviously, they don't cancel one other out within such a material. We will find out that magnetized matter is equivalent to the current distribution:

J = curl M.

Auxiliary Field H (Magnetic Intensity):

We are supposing that magnetization is due to current related by the atomic magnetic moments and spin of the electron. Such currents are termed as bound currents or magnetization amperian current. The current density Jm in equation Jm = ∇ x M is the bound current set up in the material.

Assume that you have a piece of magnetized material. Then what field does this object make? The answer is that the field generated by this object is merely the field generated by the bound currents established in it. Assume that we wind up a coil around this magnetic material and send via this coil a certain current 'I'. Then the field generated will be the sum of the field due to bound currents and the field due to current 'I'. The current 'I' is termed as the free current as it is flowing via the coil and we can assess it by joining an ammeter in series by the coil. The total current density 'J' can be represented as:

J = Jf + Jm

Here, Jf stands for the free current density 

Now, use Ampere's law to determine the field. In differential form, it is represented as:

∇ x B = µoJ

By using the equation J = Jf + Jm, Ampere's law would then take the form as:

∇ x B = µo (Jf + Jm)

Now, from the equation, Jm = ∇ x M; we have:

∇ x B = µoJf + µo (∇ x M)

∇ x [(B/µo) - M] = Jf

The above equation is the differential equation for the field [(B/µo) - M] in terms of its source Jf, the free current density. This vector is given a new symbol H, that is,

(B/µo) - M = H

The vector 'H' is termed as the magnetic intensity vector, a name which correctly belongs to 'B', however for historical reasons, has been given to 'H'. By using the two above equation, we have:

∇ x H = Jf

In another words, 'H' is associated to the free current in the way 'B' is associated to the total current, bound plus free. It can as well be written in the integral form as:

∫ H. dI = If

Here, If is the conduction current via the surface bounded through the path of the line integral on the left. Here the line integral of 'H' is around the closed path, which might or might not pass via the material. The equation can be employed to compute 'H', even in the presence of the magnetic material.

Relationship between B and H for Magnetic Material:

The magnetic fields produced by currents and computed from Ampere's Law or the Biot-Savart law is characterized through the magnetic field 'B' measured in Tesla. However if the generated fields pass via magnetic materials that themselves contribute internal magnetic fields, ambiguities can occur about what portion of the field comes from the external currents and what comes from the material itself. This has been general practice to state the other magnetic field quantity, generally termed as the magnetic field strength assigned by 'H'. It can be stated by the relationship:

H = Boo = B/μo - M

and consists of the value of unambiguously designating the driving magnetic influence from external currents in the material, independent of the  magnetic response of the material. The relationship for B can be written in the form of:

B = μo (H + M)

Here, H and M will encompass the similar units, amperes/meter. The other generally used form for the relationship between B and H is:

B = μmH

Here, μ = μm = Kmμo

Here, μo being the magnetic permeability of the space and Km the relative permeability of the material. Whenever the material doesn't respond to the external magnetic field by generating any magnetization, then Km = 1. The other generally employed magnetic quantity is the magnetic susceptibility that specifies how much the relative permeability varies from one.

Magnetic susceptibility χm = Km - 1

For paramagnetic and diamagnetic materials the relative permeability is much close to 1 and the magnetic susceptibility much close to zero. For ferromagnetic materials, such quantities might be much large.

Magnetic Circuits:

364_magnetic circuit.jpg

Magnetic circuit is a closed path to which a magnetic field, exhibited as lines of magnetic flux, is confined. In contrary to an electric circuit via which electric charge flows, nothing in reality flows in the magnetic circuit.

In a ring-shaped electromagnet having a small air gap, the magnetic field or flux is about entirely confined to the metal core and the air gap that altogether form the magnetic circuit. In an electric motor, the magnetic field is mostly confined to the magnetic pole pieces, the rotor, the air gaps among the rotor and the pole pieces and the metal frame. Each and every magnetic field line forms a complete unbroken loop. The entire lines constitute altogether the total flux. When the flux is divided, in such a manner that part of it is confined to a part of the device and part to the other, the magnetic circuit is termed as parallel. When the entire flux is confined to a single closed loop, as in a ring-shaped electromagnet, the circuit is termed as a series magnetic circuit.

In equivalence to an electric circuit in which the current, the electromotive force (or voltage), and the resistance are associated through Ohm's law (that is, current equivalents electromotive force divided by resistance), an identical relation has been developed to illustrate a magnetic circuit.

The magnetic flux is equivalent to the electric current. The magneto motive force, mmf, is equivalent to the electromotive force and might be considered the factor which sets up the flux. The mmf is equal to the number of turns of wire carrying an electric current and consists of units of ampere-turns. If either the current via a coil (that is, as in an electromagnet) or the number of turns of wire in the coil is raised, the mmf is bigger; and if the rest of the magnetic circuit remains similar, the magnetic flux rises proportionally.

The reluctance of a magnetic circuit is equivalent to the resistance of the electric circuit. Reluctance of the given portion of a magnetic circuit is proportional to its length and inversely proportional to the cross-sectional area and a magnetic property of the given material termed as its permeability. In a series magnetic circuit, the net reluctance equivalents the sum of the individual reluctances encountered about the closed flux path. In a magnetic circuit, in summary, the magnetic flux is quantitatively equivalent to the magneto motive force divided by the reluctance.

Φ = Fm/S

Here, Φ = magnetic flux in Weber (w)

Fm = m.m.f in Ampere turns (At)

S = Reluctance in Ampere turns per Weber (A/W)

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