Download 6. Magnetic Fields in Matter

6. Magnetic Fields in Matter
Matter becomes magnetized in a B field.
Induced dipoles:
Diamagnets
Permanent dipoles : Paramagnets
Ferromagnets
Magnetic dipoles are different from electric dipoles.
The dipoles are atomic current loops.
e l
l
m
   B orbital
2me 

s
m  2 B
spin

The angular momenta l and s are quantized, i. e. they take fixed
values, so does m.
Torque on a magnetic dipole (current loop):
Force on a magnetic dipole:
N  mB
F  ( m  B )
Derivation for the square loop gives the general result.
Liquid oxygen is paramagnetic. Its dipoles are pulled into
The inhomogeneous field of the permanent magnet.
Paramagnetism
The B field aligns the magnetic moment of the atoms/molecules.
The thermal motion makes the orientation random.
Competition results in partial alignment
Magnetization
m

M
i
V
Averaging over a small volume,
which contains many atomic dipoles.
Diamagnetism
The dipole moments of all atomic orbitals change, because
the orbital motion is changed.
The change m has the opposite direction of B.
Much weaker than paramagnetism. Only important, if
paramagnetism is zero.
A superconductor is a perfect diamagnet. Here, the superconducting
Pendelum bob is repelled by the permanent magnet.
Field of a Magnetized Object
We consider the macroscopic field, which is the average over
a small volume containing many dipoles.
o M(r' )  rˆ
A (r ) 
d '
2

4
r
o J b (r' )  rˆ
o K b (r' )  rˆ
A (r ) 
d ' 
d '


4
4
r
r
Bound surface current
Bound volume current
K b  M  nˆ
Jb    M
Interpretation of the surface current
Bound surface current
K b  M  nˆ
Interpretation of the bound volume current
Bound volume current
Jb    M
Example 6.1
Field of the uniformly
magnetized sphere.
The Auxiliary Field H
J  Jb  J f
H
B
o
M
H  J f
The free current is at our disposal,
the bound current is generated by
the material.
Auxiliary field
Ampere’s law
Many other authors call H “magnetic field”
and B “induction” or “flux density”.
Linear Media
For paramagnets and diamagnets there are the linear relations
M   mH
B  H
Magnetic susceptibility
m
Permeability
Permeability of free space
  o (1   m )

o
Example 6.2
Example 6.3
Solenoid filled with linear
Material.
Boundary conditions
H

above
H

below
 ( M
||
||
H above
 H below
 Kf

above
M

below
)
At surfaces between materials of different susceptibility:
 H  0
Ferromagnetism
Unlike in paramagnetic material, there is a strong interaction
between the spins of the atoms/molecules, which aligns them.
The ferromagnet is composed of domains with different orientation
of M. In the unmagnetized state they compensate each other.
Domains in an Fe-3% Si crystal observed in a scanning electron
microscope. The four colors indicate the four possible domain
directions.
In the presence of an external field the domains with an M
that is similar to H grow. Saturation is reached when only the
best domain survived.
Hysteresis loop
Magnetic field lines on a cobalt magnetic recording tape. The
Solid arrows indicate the encoded magnetic bits.