Digital Signal Processing Reference
In-Depth Information
SOLUTION According to equation (2-80), the force exerted on the loop car-
rying current
I
2
in the presence of
B
1
is
B
C
×
B
1
) dl
=
I
2
I
1
µ
0
2
π
A
a
x
dx
×
a
y
B
a
z
dz
×
a
y
F
loop
( I
2
=
x
+
b
D
C
−
A
D
−
×
a
y
×
a
y
a
+
a
x
dx
x
+
a
z
dz
Note that the segments
AB
and
CD
are equal but opposite, so they cancel.
C
A
=
I
2
I
1
µ
0
2
π
B
a
z
dz
×
a
y
D
−
a
z
dz
×
a
y
F
loop
b
+
a
From the right-hand rule, the cross products are as follows:
a
z
×
a
y
=−
a
x
−
a
z
×
a
y
=
a
x
Therefore, the force is reduced to
a
(D
−
A)
=
I
2
I
1
µ
0
2
π
1
b
(B
−
C)
+
a
x
1
F
loop
−
a
x
Since segments
BC
=
DA
, we can call this length
d
:
1
a
−
=
a
x
I
2
I
1
µ
0
d
2
π
1
b
F
loop
Therefore, the loop will be pushed away from the wire in the direction of
a
x
.
Note that the magnetic force has caused the wire loop to move. Since work
is force
×
distance, it would be easy to conclude that the magnetic force has
performed work. However, equation (2-81) explicitly states that the magnetic field
can do no work. What is performing work? To answer this question, consider
the force vector on a single segment of the loop as soon as it begins to move.
Remember that the force is perpendicular to the direction of the current flow, and
the current flow is defined by the movement of charge. When the loop moves, the
direction of the current flow
I
2
will be altered. To understand this, Figure 2-17
shows that the direction in which a single charge in the loop will travel when
the loop is moved in the
+
x
-direction. Instead of moving from right to left, it is
moved up and to the left because the loop is moving in the
+
x
-direction. This
will cause the force vector, which must remain perpendicular to the current flow,
to tilt to the right, as shown in Figure 2-17. When the force vector tilts, the
component
a
z
F
z
opposes the charge flow of the current
I
2
in the loop. For
I
2
to
remain constant, the source of the current must overcome this force.
This leads
us to the conclusion that the power source is performing the work!
The magnetic
field simply alters the direction of the force vector.
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