Lecture 12 Notes, 95.657 Electromagnetic Theory I, Fall 2011
Dr. Christopher S. Baird, UMass Lowell
1. Review of Magnetostatics in Magnetic Materials
 Currents give rise to curling magnetic fields:
∇×
B
=
0
J
total
or
∇×
H
=
J
or
∇×
M
=
J
M
where
J
total
=
J
J
M
 There are no magnetic monopoles:
∇⋅
B
=
0
which leads to
∇⋅
H
=−∇⋅
M
 Defining a vector potential
B
=∇×
A
leads to:
∇
2
A
=−
0
J
total
and
A
=
0
4
∫
J
total
x
'
∣
x
−
x
'
∣
d
x
'
 In a region where the magnetic material is linear and uniform so that
B
=
H
we can apply all of
the
B
field equations to the free current
J
instead of the total current
J
total
if we replace the
permittivity of free space
μ
0
with the permittivity of the material
μ
. For instance:
∇
2
A
=−
J
and
A
=
4
∫
J
x
'
∣
x
−
x
'
∣
d
x
'
 The boundary conditions for any type of materials are:
B
2
−
B
1
⋅
n
=
0
and
n
×
H
2
−
H
1
=
K
2. Special Cases in Magnetostatics
 If the materials are linear
and
there is no free current density in the region of space where we want
to know the fields (
J
= 0), then the equation reduces to:
∇
2
A
=
0
 These can be solved in the usual way with appropriate boundary conditions.
 An alternate approach is to define a scalar potential
B
=−∇
M
so that the zerodivergence
equation becomes:
∇
2
Ψ
M
=
0
 If there is no current density,
J
= 0, and if the material is
not
linear, but instead the magnetization
M
is known and fixed (such as in permanent magnets), the equations reduces to:
∇
2
A
=−μ
0
J
M
and
A
=
μ
0
4
π
∫
J
M
(
x
'
)
∣
x
−
x
'
∣
d
x
'
where
J
M
=∇×
M
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View Full Document The alternate scalar approach
H
=−∇
M
can also be used in this case. The statement of no
magnetic monopoles really means that the divergence of the
H
field and the
M
field are equal:
∇⋅
B
=
0
∇⋅
0
H
0
M
=
0
−∇⋅
H
=∇⋅
M
∇
2
Φ
M
=∇⋅
M
 where we can now treat the divergence of the magnetization as an effective magnetic charge
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 Fall '11
 Staff
 Current, Mass, Magnetic Field, Cos, Faraday

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