Plasma Electrodynamics

1
Magnetism in the Presence of Matter

• Magnetization
• Magnetization current density
• H-field and boundary conditions
• Linear isotropic homogeneous magnetic materials
• Magnetic energy and force
• Ferromagnetic materials

2
Atomic Current Loops and Magnetization
Amperian current
Bo
Torque Tm X Bo
Moment mIA
m
Nucleus of atom
Area A

-
I
Electron
Current I
Moment n N m
n atomic loops per large loop
Equivalent large loop
N large loops
Magnetization
3
Magnetic Dipoles and Magnetization

• Magnetization M, defined as the magnetic dipole
  moment per unit volume, analogous to the
  polarization P.
• Uniformly magnetized rod and equivalent air-core
  solenoid

K
K
A
M
B
M
B
n3
L
A
If the solenoid is the same length and diameter
as the rod, and if KK, then externally the
solenoid is the magnetic equivalent of the rod.
where n is the unit vector normal to the plane
containing the field vector.
4
Magnetization Current Densities
For the dipole moment of the volume d3r,
dmM(r)d3r
Total vector potential
or
Magnetization current densities
5
Magnetic Vectors H-field, etc
The flux density B is always the result of a
current or its equivalent.
When the vector potential A is introduced,
Where

• Magnetic flux density B, magnetic field H, and
  magnetization M are related as following
• Relativistic permeability, , and
  magnetic susceptibility from
• In non-isotropic media, such as crystal, M and H
  are, in general, not in the same direction.
  Hence, a general expression is
• For isotropic media or certain special cases in
  non-isotropic media, a more concise expression
  is possible.

The location where may be regarded as
the location of the magnetic poles of a
magnetized object, e.g. the end face of the rod.
6
Boundary Relations
The normal component of the flux density B is
continuous across the boundary between two media.
z
Medium 1, ?1
Bn1
y
?x
Medium 2, ?2
?y
Bn2
x
from
or
The change in the tangential components of H
across a boundary is equal in magnitude to the
sheet -current density K on the boundary.
?x
n
Ht1
Medium 1, ?1
Ht1
?y
Ht2
Medium 2, ?2
Ht2
n
without sheet-current
Current sheet of linear density K perpendicular
to page
7
Magnetic Energy
Magnetic energy in all spaces
Changes in magnetic energy,
Using
Magnetic energy density
For the presence of matter
8
Interaction Energy and Magnetic Force
Interaction energy of a permanent dipole in an
external field
Total interaction energy
Ex) Permeable rod in a long solenoid
Magnetic field
Energy density in the unoccupied vacuum region
Total energy
Force
In terms of magnetic pressure, outward normal
pressures corresponding to the energy density at
the boundary
Resultant force at the interface,
Tend to move the material so as to increase the
total magnetic energy of the system !
9
Magnetic Classification of Materials

• Nonmagnetic vacuum
• Diamagnetic Weakly magnetic, An applied
  magnetic field causes the spin moment to slightly
  exceed the orbital moment, resulting in a small
  net magnetic moment which opposes the applied
  field B. repelled by a bar magnet.
• Example Bismuth (Faraday 1846)
• Paramagnetic Significant magnetism. Attracted
  to a bar magnet.
• Example Aluminum
• Ferromagnetic Strongly magnetic (atomic
  moments aligned). Attracted to a bar magnet.
  Becomes paramagnetic above Curie temperature.
• Examples Iron, nickel, cobalt
• Antiferromagnetic Nonmagnetic even in presence
  of applied magnetic field. Moments of adjacent
  atoms align in opposite direction
• Example Manganese oxide (MnO2)
• Ferrimagnetic less magnetic than ferromagnetic
  material.
• Example Iron ferrite
• Ferrite Ferrimagnetic material with low
  electrical conductivity. Useful as inductor cores
  for an ac applications because of less eddy
  currents and ohmic losses.
• Superparamagnetic Ferromagnetic materials
  suspended in dielectric matrix, exchange forces
  cannot penetrate to adjacent particles. Used in
  audio and video tapes.

10
Relative Permeablity

• Nonmagnetic 1. Permeability of ?o4? x10-7
• Diamagnetic slightly less than 1. Independent
  of the applied magnetic field
• Paramagnetic slightly greater than 1 .
  Independent of the applied magnetic field
• Ferromagnetic much greater than 1. A wide
  range for different applied fields even with
  Hysteresis.

Substance Group type Relative permeability, ?r
Bismuth Diamagnetic 0.99983 Silver Diamagnetic
0.99998 Lead Diamagnetic 0.999983 Copper Diam
agnetic 0.999991 Water Diamagnetic 0.999991 Air
Paramagnetic 1.0000004 Aluminum Paramagnetic 1
.00002 Palladium Paramagnetic 1.0008 Cobalt Fe
rromagnetic 250 Nickel Ferromagnetic 600 Feroxcu
be 3(Mn-Zn-ferrite powder) Ferrimagnetic 1,500 Mil
d steel (0.2 C) Ferromagnetic 2,000 Iron (0.2
impurity) Ferromagnetic 5,000 Purified iron
(0.05 impurity) Ferromagnetic 200,000 Superalloy
(5 Mo, 79 Ni) Ferromagnetic 1,000,000
11
Ferromagnetism

• The atomic dipoles tend to align in the same
  direction over regions, called domains.
• When all the domains are in the same direction
  by applying strong external field, magnetic
  saturation is reached. The crystal is then
  magnetized to the maximum extent.
• If the majority of the domains retain their
  directions after the applied field is removed,
  the specimen is said to be permanently
  magnetized.
• If the temperature is raised above Curie point,
  the substance changes from ferromagnetic to
  paramagnetic.
• Induced magnetization Magnetization which
  appears only in the presence of an applied
  magnetic field.
• Magnetization curve and Hysteresis

Flux density, B
Bm
Br
Initial magnetization curve
Residual flux density or remnance
BH
Demagnetization curve
-Hc
Hm
-Hm
Hc
Energy product or Stored energy density in the
magnet
Magnetic field, H
Coercive force
-Br
Hysteresis loop
-Bm
0