Electromagnetics Workshop – Week 13
Solutions
Magnetostatics
1
Problem 1:
A long solenoid of length
L
is wound with many layers of thin wire. A cross section of the
solenoid is shown in the figure. There are a total of
N
turns, each carrying a current
I
. The inner
radius of the solenoid is
a
and the outer radius is
b
. Neglect fringing of the fields near the end of
the solenoid.
a)
What is the magnetic field
H
in the region
b
ρ
>
?
b)
Find the magnetic field
H
in the region
a
ρ
<
.
c)
Find the magnetic field
H
in the region
a
b
ρ
<
<
.
Problem 1 Solution:
.
encl
d
I
⋅
=
∫
H
A
v
(a)
b
ρ
>
:
.
0
0
encl
I
=
⇒
=
H
(b)
a
ρ
<
:
.
encl
z
z
N I
I
N I
H L
N I
H
L
=
⇒
=
⇒
=
[A/m]
(c)
a
b
ρ
<
<
:
At
b
ρ
=
:
.
0
0
encl
I
=
⇒
=
H
At
a
ρ
=
:
.
encl
z
z
N I
I
N I
H L
N I
H
L
=
⇒
=
⇒
=
[A/m]
The field should vary smoothly between these two points.
.
encl
z
z
b
b
N I
b
I
N I
H L
N I
H
a
b
a
b
L
a
b
ρ
ρ
ρ
−
−
−
⎛
⎞
⎛
⎞
⎛
⎞
=
⇒
=
⇒
=
⎜
⎟
⎜
⎟
⎜
⎟
−
−
−
⎝
⎠
⎝
⎠
⎝
⎠
[A/m]

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Electromagnetics Workshop – Week 13
Solutions
Magnetostatics
2
Problem 2:
Determine the magnetic field
H
at the point (
x
, 0, 0) due to the infinitely long wire with a right-
angle bend, as shown below (the wire goes to infinity in the –
x
and –
y
directions).
Helpful integral:
(
)
3 2
2
2
2
2
2
1
du
u
a
u
a
u
a
=
+
+
∫
Problem 2 Solution:
This is the superposition the magnetic field of two semi-infinite wires.
“Wire 1” is the wire on
the
x
-axis.
“Wire 2” is the wire on the
y
-axis.
Wire 1:
(
)
ˆ
x
x
′
=
−
1
R
x
(
)
(
)
2
ˆ
ˆ
0
4
C
I
dx
x
x
π
×
′
=
=
′
−
∫
1
x
x
H
Wire 2:
2
ˆ
ˆ
x
y
′
=
−
R
x
y
(
)
(
)
(
)
(
)
(
)
0
0
0
2
3 2
3 2
2
2
2
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
4