Magnetic Field caused by Polarization of Magnetic Material
An object with a magnetic dipole moment, such as a bar magnet, produces a familiar magnetic field in the space surrounding the object. The field is called the dipole magnetic field. In this section we will calculate the magnetic field inside a polarized magnetic material.
Suppose that we have a large cylindrical sample of magnetic material. We assume that any small chunk we carve out of the interior of the sample will have a dipole moment, just as in the case of dielectric materials.
We make another important assumption: The dipole moment of a small chunk of material is associated with a current circulating in the chunk. This is plausible for magnetic moments due to electrons orbiting around a nucleus in an atom - the "orbital magnetic moment" - but is evidently incorrect for the magnetic moment of an electron itself: There is no evidence of currents within the electron (the radius of an electron appears to be zero). Nonetheless, our results agree with experiment.
For simplicity we suppose that the current circulates around the exterior of the chunk. We will calculate the effective current associated with all of the chunks in the large sample, and use this current to calculate the magnetic field inside of the large sample.
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|
| Dipole moment associated with a current loop. |
We can relate the magnetic dipole moment of a chunk to the current which circulates around it by recalling the dipole moment of a loop of current-carrying wire:
Experiments with macroscopic current carrying wires, placed in a uniform magnetic field, show that the dipole moment of a single loop of current-carrying wire is
where 
is the current in the wire and 
is the area of the loop, as in the figure.
The direction of the dipole moment is related to the direction of the current by a right hand rule.
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| Dipoles clustered in a plane are equivalent to a |
| single dipole. |
We next pack a group of identical dipole chunks into a circle and ask how their currents combine to produce a net magnetic dipole moment. We assume that all the dipoles are aligned in the same direction, as shown in the figure.
Within the material the net current is zero. (Where two chunks touch each other, their currents flow in opposite directions.) The only non-zero current is around the outside edge, where no cancellation occurs.
At any point on the edge, the current is
, the same as in an individual chunk. The dipole moment of the large circle is
The additivity of the currents leads to additivity of the dipole moments.
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| A solid polarized paramagnetic sample, of N layers, |
| each layer carrying an effective current . |
Now stack these circles in layers to make a solid cylindrical volume, as in the figure. This puts us in a position to calculate the magnetic (
) field within the cylinder, caused by the dipole moments.
Because the effective current flows around the outside of the cylinder, we can use the formula for the magnetic field of a solenoid to calculate
:
where
is the current. In terms of the figure,
Using the relation between current and magnetic moment of one layer, we can write
.
The term in braces is the (average) dipole moment per unit volume, which we call
, the magnetic polarization. We can write the field due to polarization of a magnetic material as:
Note that the
field is in the same direction as the field. Remember that this is the magnetic field inside the material. Since the polarization aligns parallel to the applied magnetic field, this means that the polarization enhances the applied magnetic field, both inside and outside of the magnetic object.
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