An electron moving in an orbit around a nucleus produces an average current along its orbit. As a consequence we can associate a magnetic moment with the orbiting electron. Suppose the electron is moving with a velocity v in an orbit with radius r. The period of this motion is equal to

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During one period T the charge e will pass each given point on the orbit. The current associated with this orbit is therefore equal to

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The magnetic moment associated with this current is equal to

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It is common to specify the orbit of an electron in terms of its angular momentum L. Using the definition of the angular momentum L we can relate the electron velocity v and the radius of its orbit r to the angular momentum L in the following manner:

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where m is the mass of the electron. Using eq.(33.4) we can express the magnetic moment of the electron in terms of the angular momentum L:

_{
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The magnetic moment of the electron is thus proportional the angular momentum
L. The angular momentum of the electron is quantized, and the only possible
values are n~~h~~ where n is an integer (n = 0, 1, 2, 3, ...) and
~~h~~ is a constant (~~h~~ = 1.06 ^{.}
10^{ -34} J s). The magnetic moment of an electron with angular
momentum L = 1 is equal to

_{
}

Since this magnetic moment is associated with the orbital motion of the
electron around the nucleus it is called the ** orbital magnetic
moment**. Another contribution to the magnetic moment is due to the
rotational motion of the electron. Classically we can regard an electron as a
small ball of negative charge spinning around its axis. The intrinsic angular
momentum, generated by the electron spin, is equal to

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This constant is also called the ** Bohr magneton**.

The total magnetic moment of an atom is equal to the vector sum of the orbital magnetic moments and the intrinsic magnetic moments of all its electrons. The contribution of the nuclear magnetic moment is small and often can be neglected. Each atom acts like a magnetic dipole and produces a small, but measurable magnetic field.

a) Two electrons are separated by a distance of 1.0 x 10^{ -10}
m. The first electron is on the axis of spin of the second. What is the
magnetic field that the magnetic moment of the second electron produces at the
position of the first ?

b) The potential energy of the magnetic moment of the first electron in this magnetic field depends on the orientation of the electrons. What is the potential energy if the spins of the two electrons are parallel ? If anti parallel ? Which orientation has the least energy ?

a) Figure 33.1 shows the orientation of the two electrons. The z-axis is
defined to coincide with the spin of electron 2. With each spinning electron a
magnetic dipole moment can be associated. Due to the negative charge of the
electron the dipole moment is pointed in a direction opposite to that of the
spin of the electron and it has a magnitude equal to u_{spin} = 9.27 x
10^{ -24} Am^{2}. The magnetic field generated by electron
will be the magnetic field generated by a dipole with dipole moment
u_{spin}. On the z-axis (and for z >> R_{e}) the field
strength will fall of as 1/z^{3} and at z = 1.0 x 10^{ -10} m
has a strength equal to

_{
}

_{
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where [theta] is the angle between the dipole moment and the magnetic field. The potential energy of electron 1 reaches a maximum value when [theta] = [pi] and a minimum value when [theta] = 0. Evaluating equation (33.9) for these extreme cases yields

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The potential energy will have a minimum value when the spins are parallel and a maximum value when the spins are anti-parallel.

Even though each atom in a material can have a magnetic moment, the direction
of each dipole is randomly oriented and their magnetic fields average to zero.
If the material is immersed in an external magnetic field, the dipoles will
tend to align themselves with the field in order to minimize the torque exerted
on them by the external magnetic field. The atoms in the material will produce
an extra magnetic field in its interior that has the same direction as the
external magnetic field. This increase in strength of the magnetic field can
be quantified in terms of the ** permeability constant**
[kappa]

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where B_{free} is the external magnetic field. The definition of the
permeability constant (see eq.(33.11)) shows that [kappa]_{m} >= 1.
For all paramagnetic materials the permeability constant is very close to 1,
and as a consequence, the increase in the magnetic field strength is rather
small.

Show that the self-inductance per unit length of a very long solenoid
filled with a paramagnetic material is equal to [kappa]_{m}
u_{0} n^{2} [pi] R^{2}, where n is the number of turns
of wire per unit length and R is the radius of the solenoid.

The magnetic field in an empty solenoid is equal to

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When the solenoid is filled with a paramagnetic material the strength of the magnetic field will increase (see eq.(33.11)) and will be equal to

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The magnetic flux enclosed by a section of the solenoid of unit length is equal to

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The change in enclosed magnetic can be obtained from eq.(33.14) by differentiating both sides with respect to time

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The emf induced by this change in magnetic flux can be calculated using Faraday's law

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From eq.(33.16) we conclude that the self-inductance of the solenoid filled with a paramagnetic material is equal to

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The alignment of the spins of some of the electrons in a ferromagnetic
material will increase the magnetic field in this material in much the same way
as the alignment of the orbital magnetic dipole moments of atoms increases the
field strength in a paramagnetic material. In a ferromagnetic material the
degree of alignment of the electron spins between neighboring atoms is high as
a result of a special force that tends to lock the spins of these electrons in
a parallel direction. This force is so strong that the spins remain aligned
even when the external magnetic field is removed. Materials with such
properties are called ** permanent magnets**. The force that is
responsible for the alignment of the electron spins occurs in only five
elements:

- Iron
- Nickel
- Cobalt
- Dysprosium
- Gadolinium

The degree of alignment of the spins in a ferromagnetic material after the
external magnetic field has been removed depends on the temperature. An
increase in the temperature of the material will increase the chance of random
rearrangement of the magnetic dipoles. Above a certain temperature, called the
** Curie temperature**, the magnetism of the ferromagnet disappears
completely.

Under conditions of maximum magnetization, the dipole moment per unit
volume of cobalt is 1.5 x 10^{ 5} Am^{2}/m^{3}.
Assuming that this magnetization is due to completely aligned electrons, how
many such electrons are there per unit volume ? How many aligned electrons are
there per atom ? The density of cobalt is 8.9 x 10^{ 3} kg and the
atomic mass is 58.9 g/mole.

The dipole moment of 1 m^{3} of cobalt is equal to 1.5 x 10^{
5} Am^{2}. Each aligned electron contributes a dipole moment of
9.27 x 10^{ -24} Am^{2}. The number of aligned electrons is
thus equal to

_{
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The number of atoms in 1 m^{3} of cobalt is equal to

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Comparing eq.(33.18) and eq.(33.19) we conclude that the total number of aligned electrons per atom is equal to 0.18.

In a diamagnetic material the magnetization arises from induced magnetic dipoles. This in contrast to the paramagnetic and ferromagnetic materials where the magnetic properties are determined by the alignment of permanent magnetic dipoles. In a diamagnetic material the dipole moments of the atoms do not align themselves with the magnetic field, but their strength is changed by the external field.

_{
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When the magnetic field is turned on, the electron will experience in addition to the electric force a magnetic force equal to

_{
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This force is radially directed (inwards or outwards depending on the direction
of v_{0} with respect to B). The condition for uniform circular motion
is now

_{
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Assuming that the size of the orbit of the electron does not change, we conclude that the effect of the magnetic field is a change in the velocity of the electron. Combining eq.(33.22) and eq.(33.20) we obtain

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It is convenient to express the velocity of the electron in terms of its angular frequency

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Substituting the expressions for v_{0} and v from eq.(33.24) into
eq.(33.23) we obtain

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or

_{
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The change in frequency [Delta][omega] is defined via the following relation

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Substituting eq.(33.27) into eq.(33.26) we obtain

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or

_{
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This frequency is called the ** Larmor frequency**, and indicates the
maximum change in the velocity of the orbital electrons when an external
magnetic field is applied. As a result of the change in the velocity of the
orbital electrons there will be a change in the orbital magnetic moment. The
orbital magnetic moment before the external magnetic field is applied can be
calculated using eq.(33.3)

_{
}

When an external magnetic field is applied the angular frequency [omega] will change by [Delta][omega]. Equation (33.30) shows that the associated change in the orbital magnetic moment is equal to

_{
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The change of the angular frequency can be positive or negative, depending on
the direction of v_{0} with respect to the direction B. The angular
frequency will increase if the direction of the field and the electron velocity
are ** not** related via the right-hand rule (field into paper in
Figure 33.2). The change in the magnetic moment of the dipole is such that the
net field strength (external + orbital magnetic field) is lowered. This means
an increase in the orbital magnetic moment if the orbital magnetic field is
directed in a direction opposite to the external field, and a decrease in the
orbital magnetic moment if the orbital magnetic field is directed in the same
direction as the external field. In both cases the total magnetic field
strength is reduced, and, consequently, the permeability constant
[kappa]

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