Paramagnetic Materials

For simplicity, we assume our sample is a long, cylindrical piece that completely fills the interior of a long, tightly wound solenoid. When there is no current in the solenoid, the magnetic dipoles in the sample are randomly oriented and produce no net magnetic field. With a solenoid current, the magnetic field due to the solenoid exerts a torque on the dipoles that tends to align them with the field. In competition with the aligning torque are thermal collisions that tend to randomize the orientations of the dipoles. The relative importance of these two competing processes can be estimated by comparing the energies involved, the energy difference between a magnetic dipole aligned with and against a magnetic field is UB=2μB

UB=2μB. If μ=9.3×10−24A⋅m2

μ=9.3×10−24A⋅m2 (the value of atomic hydrogen) and B = 1.0 T, then

At a room temperature of 27oC

27oC the thermal energy per atom is

which is about 220 times greater than UB

UB. Clearly, energy exchanges in thermal collisions can seriously interfere with the alignment of the magnetic dipoles. As a result, only a small fraction of the dipoles is aligned at any instant.

The four sketches of Figure 12.7.1

12.7.1 furnish a simple model of this alignment process. In part (a), before the field of the solenoid (not shown) containing the paramagnetic sample is applied, the magnetic dipoles are randomly oriented and there is no net magnetic dipole moment associated with the material. With the introduction of the field, a partial alignment of the dipoles takes place, as depicted in part (b). The component of the net magnetic dipole moment that is perpendicular to the field vanishes. We may then represent the sample by part (c), which shows a collection of magnetic dipoles completely aligned with the field. By treating these dipoles as current loops, we can picture the dipole alignment as equivalent to a current around the surface of the material, as in part (d). This fictitious surface current produces its own magnetic field, which enhances the field of the solenoid.

 The alignment process in a paramagnetic material filling a solenoid (not shown). (a) Without an applied field, the magnetic dipoles are randomly oriented. (b) With a field, partial alignment occurs. (c) An equivalent representation of part (b). (d) The internal currents cancel, leaving an effective surface current that produces a magnetic field similar to that of a finite solenoid.

We can express the total magnetic field B⃗ 

B→ in the material as

where B⃗ 0

B→0 is the field due to the current I0

I0 in the solenoid and B⃗ m

B→m is the field due to the surface current Im

Im around the sample. Now B⃗ m

B→m is usually proportional to B⃗ 0

B→0 a fact we express by

where χ

χ is a dimensionless quantity called the magnetic susceptibility. Values of χ

χ for some paramagnetic materials are given in Table 12.7.2

12.7.2. Since the alignment of magnetic dipoles is so weak, χ

χ is very small for paramagnetic materials. By combining Equation

12.7.3 and Equation

12.7.4, we obtain:

For a sample within an infinite solenoid, this becomes

This expression tells us that the insertion of a paramagnetic material into a solenoid increases the field by a factor of (1+χ)

(1+χ). However, since χ

χ is so small, the field isn’t enhanced very much.

The quantity

is called the magnetic permeability of a material. In terms of μ

μ, Equation 12.7.6

12.7.6 can be written as

for the filled solenoid.

Diamagnetic Materials

A magnetic field always induces a magnetic dipole in an atom. This induced dipole points opposite to the applied field, so its magnetic field is also directed opposite to the applied field. In paramagnetic and ferromagnetic materials, the induced magnetic dipole is masked by much stronger permanent magnetic dipoles of the atoms. However, in diamagnetic materials, whose atoms have no permanent magnetic dipole moments, the effect of the induced dipole is observable.

We can now describe the magnetic effects of diamagnetic materials with the same model developed for paramagnetic materials. In this case, however, the fictitious surface current flows opposite to the solenoid current, and the magnetic susceptibility χ

χ is negative. Values of χ

χ for some diamagnetic materials are also given in Table 12.7.2

12.7.2.

Ferromagnetic Materials

Common magnets are made of a ferromagnetic material such as iron or one of its alloys. Experiments reveal that a ferromagnetic material consists of tiny regions known as magnetic domains. Their volumes typically range from 10−12

10−12 to 10−8m3

10−8m3, and they contain about 1017

1017 to 1021

1021 atoms. Within a domain, the magnetic dipoles are rigidly aligned in the same direction by coupling among the atoms. This coupling, which is due to quantum mechanical effects, is so strong that even thermal agitation at room temperature cannot break it. The result is that each domain has a net dipole moment. Some materials have weaker coupling and are ferromagnetic only at lower temperatures.

If the domains in a ferromagnetic sample are randomly oriented

, the sample has no net magnetic dipole moment and is said to be unmagnetized. Suppose that we fill the volume of a solenoid with an unmagnetized ferromagnetic sample. When the magnetic field B⃗ 0

B→0 of the solenoid is turned on, the dipole moments of the domains rotate so that they align somewhat with the field, as depicted in Figure 12.7.1b

12.7.1b. In addition, the aligned domains tend to increase in size at the expense of unaligned ones. The net effect of these two processes is the creation of a net magnetic dipole moment for the ferromagnet that is directed along the applied magnetic field. This net magnetic dipole moment is much larger than that of a paramagnetic sample, and the domains, with their large numbers of atoms, do not become misaligned by thermal agitation. Consequently, the field due to the alignment of the domains is quite large.

: (a) Domains are randomly oriented in an unmagnetized ferromagnetic sample such as iron. The arrows represent the orientations of the magnetic dipoles within the domains. (b) In an applied magnetic field, the domains align somewhat with the field. (c) The domains of a single crystal of nickel. The white lines show the boundaries of the domains. These lines are produced by iron oxide powder sprinkled on the crystal.

Besides iron, only four elements contain the magnetic domains needed to exhibit ferromagnetic behavior: cobalt, nickel, gadolinium, and dysprosium. Many alloys of these elements are also ferromagnetic. Ferromagnetic materials can be described using Equation 

through Equation the paramagnetic equations. However, the value of χ

χ for ferromagnetic material is usually on the order of 103

103 to 104

104, and it also depends on the history of the magnetic field to which the material has been subject. A typical plot of B (the total field in the material) versus B0

B0 (the applied field) for an initially unmagnetized piece of iron is shown in

. Some sample numbers are (1) for B0=1.0×10−4T

B0=1.0×10−4T, B=0.60 T

B=0.60 T, and χ=(0.60/1.0×10−4)−1≈6.0×103

χ=(0.60/1.0×10−4)−1≈6.0×103; for (2) for B0=6.0×10−4T

B0=6.0×10−4T, B=1.5 T

B=1.5 T, and χ=(1.5/6.0×10−4)−1≈2.5×103

χ=(1.5/6.0×10−4)−1≈2.5×103.

: (a) The magnetic field B in annealed iron as a function of the applied field B0

B0.

When B0

B0 is varied over a range of positive and negative values, B is found to behave as shown in Figure 12.7.3

12.7.3. Note that the same B0

B0 (corresponding to the same current in the solenoid) can produce different values of B in the material. The magnetic field B produced in a ferromagnetic material by an applied field B0

B0 depends on the magnetic history of the material. This effect is called hysteresis, and the curve of

is called a hysteresis loop. Notice that B does not disappear when B0=0

B0=0 (i.e., when the current in the solenoid is turned off). The iron stays magnetized, which means that it has become a permanent magnet.

Figure 12.7.4

12.7.4: A typical hysteresis loop for a ferromagnet. When the material is first magnetized, it follows a curve from 0 to a. When B0

B0 is reversed, it takes the path shown from a to b. If B0

B0 is reversed again, the material follows the curve from b to a.

Like the paramagnetic sample

, the partial alignment of the domains in a ferromagnet is equivalent to a current flowing around the surface. A bar magnet can therefore be pictured as a tightly wound solenoid with a large current circulating through its coils (the surface current).

that this model fits quite well. The fields of the bar magnet and the finite solenoid are strikingly similar. The figure also shows how the poles of the bar magnet are identified. To form closed loops, the field lines outside the magnet leave the north (N) pole and enter the south (S) pole, whereas inside the magnet, they leave S and enter N.

: Comparison of the magnetic fields of a finite solenoid and a bar magnet.

Ferromagnetic materials are found in computer hard disk drives and permanent data storage devices (Figure 12.7.6

12.7.). A material used in your hard disk drives is called a spin valve, which has alternating layers of ferromagnetic (aligning with the external magnetic field) and antiferromagnetic (each atom is aligned opposite to the next) metals. It was observed that a significant change in resistance was discovered based on whether an applied magnetic field was on the spin valve or not. This large change in resistance creates a quick and consistent way for recording or reading information by an applied current.

Figure 12.7.6

12.7.6: The inside of a hard disk drive. The silver disk contains the information, whereas the thin stylus on top of the disk reads and writes information to the disk.

Earth’s Magnetism

On the surface of the earth, the strength of the earth’s magnetic field varies from place to place and is of order 10^-5Testla.

Dynamo effect

Though there are many theories on earth’s magnetic field, dynamo effect seems to be most accepted one.

• The earth consists of core, mantle and crust
• In the outer core, there is molten iron and nickel
• The convective motion of these metallic fluids, results in electrical currents
• The magnetic field is due to this electrical currents

Magnetic lines of the earth

Let us imagine a magnetic dipole is present in the centre of the earth. Now, draw magnetic field lines. The magnetic lines of the earth resemble the same.

Magnetic poles :

The one small difference is the axis of the dipole does not coincide with the axis of rotation of earth and is tilted by approximately 11.3⁰

Hence, apart from the geographic North Pole N_g and South Pole S_g, we also have magnetic North pole N_m and magnetic South Pole S_m.

Nomenclature of poles :

We know that in the case of a bar magnet, the magnetic lines go from the South Pole to the North Pole.  So field lines come out from the North Pole.

However, in the case of the earth, the field lines go into the earth at the magnetic North pole and come out at the magnetic South pole.

When magnetic needle was suspended freely, the north pole of the needle coincided with the magnetic north pole. Hence, it became conventional to call it as North magnetic meridian, which is close to Geographic North pole.

Thus, in reality, the north magnetic pole behaves like South pole of bar magnet and vice versa.This is the reason why the needle having North pole is attached by the south pole behaviour of earth, though called as north magnetic pole.

Elements of earth’s magnetic field

The 3 elements of earth’s magnetic field are – (1) Angle of declination (α)

(2) Angle of dip (δ)and (3) Horizontal component of earth’s magnetic field (H_e)

To describe the magnetic field of earth at any point on the surface of the earth, we require the above 3 quantities.

Angle of declination (α) :

The angle between the geographical meridian and magnetic meridian is known as angle of declination.

This angle is smaller at the equator and greater at higher latitudes. At equator, the magnetic meridian will be close to geographic meridian.

Angle of dip / inclination (δ) :

The angle made between the total of earth’s magnetic field (B_e) with the surface of the earth (horizontal component) in the magnetic meridian is known as angle of dip.

The yellow lines indicate the surface of the earth. The blue lines indicate the total magnetic field of earth which is actually tangent to the magnetic field line. This angle of dip is 90 degrees at the equator and 0 degrees at the poles.

Horizontal component of earth’s magnetic field (H_{e) :}

The total magnetic field on the surface of the earth can be resolved into horizontal component H_e and vertical component Z_e. The angle made by Be with He is the angle of dip δ

Hence,H_{e =}  B_e cos δ

V_{e =}  B_e sin δ

V_e  / H_{e =}  B_e tan δ

Numericals: A magnetic needle free to rotate in a vertical plane parallel to the magnetic meridian has its north tip pointing down at 22⁰ with the horizontal. The horizontal component of earth’s magnetic field at the place is known to be 0.35 G.

Given H = 0.35G ,             δ = 22^{0  ,}            H = B_e cos δ

B_e = H / cos 22 = 0.35/ cos 22 = 0.38G