Biomedical Engineering Reference
In-Depth Information
z
revolving
permanent
magnet
South
ω
y
B
0
North
d
q
dS
x
FIGURE 16.29
Voltage induced in wire loop of radius
d
by a revolving permanent magnet of strength B
0
.
This overall result implies that an orbiting charge can act like a small magnet with its own
north and south pole.
Can a moving magnetic field create a current or voltage in a wire? For the third case,
consider the arrangement in Figure 16.29, where a wire loop of radius
d
is perpendicular
to the
x
-axis and where a permanent magnet of strength
B
0
is whirling about the
z
-axis at
a constant angular frequency
o
. The angle between the
x
-axisandthemagnetaxiscanbe
described as
. Then, Faraday's law specifies that the voltage created in the loop by
the spinning magnet can be written in terms of the electric field
y ¼ o
t
E
around the loop. The
2
, and its vector is perpendicular to the loop. If the field rotating
area of the loop is
S
¼ p
d
relative to the
x
-axis is B
0
cos
o
t
and has a direction along vector
B
, then according to
Faraday's law,
V
¼
d
2
cos
2
sin
dt
ð
B
0
p
d
o
t
Þ¼o
B
0
p
d
o
t
ð
16
:
51
Þ
The voltage picked up from the rotating magnet is sinusoidal and is maximum when the
axis of the magnet is perpendicular to the plane of the loop and is zero when the axis is
parallel.
For the fourth case, a whirling charge is placed in a strong static magnetic field
B
0
,as
shown in Figure 16.30. Here, the action of the field on the charge exerts a force on the
charge described by the Lorentz force equation
F
¼
qv
B
,
or
j
F
j¼
q
j
v
jj
B
j
sin
y
ð
16
:
52a
Þ
where this vector cross-product notation means the velocity vector
is tangential to the
orbit at the position of the charge and the force is exerted outward perpendicular to both
v
v
and the applied field direction,
B
0
and the angle between
and
B
0
,
90
. The magni-
v
y ¼
tude of this force can be rewritten for this case as
y ¼
mv
2
=
r
j
F
j¼
q
j
v
jj
B
j
sin
ð
16
:
52b
Þ
