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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Lecture 10
Perfect Metals in Magnetism and Inductance
In this lecture you will learn:
• Some more about the vector potential
• Magnetic field boundary conditions
• The behavior of perfect metals towards timevarying magnetic fields
• Image currents and magnetic diffusion
• Inductance
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
The Vector Potential  Review
0
.
=
∇
A
r
In
electroquasistatics
we had:
0
=
×
∇
E
r
Therefore we could represent the Efield by the scalar potential:
φ
−∇
=
E
r
In
magnetoquasistatics
we have:
Therefore we can represent the Bfield by the vector potential:
( ) ( )
0
.
.
=
∇
=
∇
H
B
o
r
r
µ
A
H
B
o
r
r
r
×
∇
=
=
A vector field can be specified (up to a constant) by specifying its curl and its
divergence
Our definition of the vector potential
is not yet unique – we have only
specified its curl
For simplicity we fix the divergence of the vector potential
to be zero:
A
r
A
r
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ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Magnetic Flux and Vector Potential Line Integral  Review
The magnetic flux
λ
through a surface is the surface integral of the Bfield
through the surface
∫∫
=
∫∫
=
a
d
H
a
d
B
o
r
r
r
r
.
.
µ
λ
Since:
A
H
B
o
r
r
r
×
∇
=
=
()
∫
=
∫∫
×
∇
=
∫∫
=
s
d
A
a
d
A
a
d
B
r
r
r
r
r
r
.
.
.
We get:
Bfield
The magnetic flux through a surface is equal to the lineintegral of the vector
potential along a closed contour bounding that surface
s
d
r
A closed contour
Stoke’s Theorem
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Vector Potential of a LineCurrent
z
I
ˆ
x
y
Consider an infinitely long linecurrent with
current
I
in the +
z
direction
The
Hfield has only a
φ
component
r
Using Ampere’s Law:
( )
I
r
H
=
π
2
Work in cylindrical
coordinates
r
I
H
2
=
⇒
But
( )
( )
r
I
r
r
A
r
r
A
A
H
o
z
z
o
o
2
1
−
=
∂
∂
⇒
∂
∂
−
=
×
∇
=
r
() ()
⎟
⎠
⎞
⎜
⎝
⎛
=
−
r
r
I
r
A
r
A
o
o
o
z
z
ln
2
If the current has only a zcomponent then the vector potential also only has a
z
component which, by symmetry, is only a function of the distance from the
linecurrent
Integrating from
r
o
to
r
:
J
A
o
r
r
−
=
∇
2
3
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Vector Potential of a LineCurrent Dipole
z
I
ˆ
+
x
y
d
+
r
−
r
Consider two infinitely long equal and
opposite linecurrents, as shown
The vector potential can be written as a
sum using superposition:
()
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
+
−
−
+
r
r
I
r
r
I
r
r
I
r
A
o
o
o
o
o
z
ln
2
ln
2
ln
2
π
µ
r
The final answer does not
depend on the parameter
r
o
r
r
Question: where is the zero of the vector potential?
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This note was uploaded on 02/02/2008 for the course ECE 3030 taught by Professor Rana during the Fall '06 term at Cornell University (Engineering School).
 Fall '06
 RANA
 Electromagnet

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