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ECE 303: Electromagnetic Fields and Waves
Fall 2007
Solutions for Prelim 1
Reminder:
Date: Tuesday 9/25
Problem 1:
A current loop above a perfect conductor.
a.
The image current is a loop of radius
a
, located at
z
=

d
, and of value

I
(
t
) (i.e., opposite direction).
b.
The
z
component of the magnetic ﬁeld due to the original loop is
H
z
(
partial
) =
I
(
t
)
4
π
±
2
πa
a
2
+ (
L

d
)
2
¶
sin
α
=
I
(
t
)
2
ˆ
a
2
[
a
2
+ (
L

d
)
2
]
3
/
2
!
Then the total ﬁeld is the superposition of the ﬁelds produced by each current loop:
H
z
=
a
2
I
(
t
)
2
ˆ
1
[
a
2
+ (
L

d
)
2
]
3
/
2

1
[
a
2
+ (
L
+
d
)
2
]
3
/
2
!
c.
By symmetry, the
x
component of the ﬁeld is zero.
Problem 2:
Finding the electric ﬁeld with a volume charge and dielectric material.
a.
The boundary condition at
x
=
d
is
ε
0
E
x
(
x
=
d
+
) =
ε
1
E
x
(
x
=
d

) where
E
x
(
x
=
d
+
) =
E
0
. So,
E
x
(
x
=
d

) =
ε
0
ε
1
E
0
. The diﬀerential form of Gauss’ Law in one dimension is
∂
∂x
(
ε
1
E
x
(
x
)) =
ρ
(
x
)
for 0
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