Physics For Scientists And Engineers 6E - part 242
Problems
965
B (T)
B
0
(T)
0.2
4.8 & 10
'
5
0.4
7.0 & 10
'
5
0.6
8.8 & 10
'
5
0.8
1.2 & 10
'
4
1.0
1.8 & 10
'
4
1.2
3.1 & 10
'
4
1.4
8.7 & 10
'
4
1.6
3.4 & 10
'
3
1.8
1.2 & 10
'
1
Table P30.70
Figure P30.69
Figure P30.67
x
P
I
L
L
r = e
θ
y
x
r
dr
d s
θ
rˆ
= /4
π
β
I
I
The loop has a length L, radius R, and carries a current I
2
.
The axis of the loop coincides with the wire. Calculate the
force exerted on the loop.
66. Measurements of the magnetic field of a large tornado
were made at the Geophysical Observatory in Tulsa,
Oklahoma, in 1962. The tornado’s field was measured to
be B " 1.50 & 10
'
8
T pointing north when the tornado
was 9.00 km east of the observatory. What current was
carried up or down the funnel of the tornado, modeled as
a long straight wire?
A wire is formed into the shape of a square of edge length
L (Fig. P30.67). Show that when the current in the loop is
I, the magnetic field at point P, a distance x from the
center of the square along its axis is
B "
#
0
IL
2
2$(x
2
*
L
2
/4)
√
x
2
*
L
2
/2
67.
68.
The force on a magnetic dipole " aligned with a nonuni-
form magnetic field in the x direction is given by
F
x
"
"""dB/dx. Suppose that two flat loops of wire each
have radius R and carry current I. (a) The loops
are arranged coaxially and separated by a variable
distance x, large compared to R. Show that the magnetic
force between them varies as 1/x
4
. (b) Evaluate the
magnitude of this force if I " 10.0 A, R " 0.500 cm, and
x " 5.00 cm.
69.
A wire carrying a current I is bent into the shape of an
exponential spiral, r " e
!
, from ! " 0 to ! " 2$ as sug-
gested in Figure P30.69. To complete a loop, the ends of
the spiral are connected by a straight wire along the x axis.
Find the magnitude and direction of
B at the origin.
Suggestions: Use the Biot–Savart law. The angle 9 between a
radial line and its tangent line at any point on the curve
r " f (!) is related to the function in the following way:
Thus in this case r " e
!
, tan 9 " 1 and 9 " $/4.
Therefore, the angle between d
s and ˆr is $ ' 9 " 3$/4.
Also
ds "
dr
sin($/4)
"
√
2 dr
tan 9 "
r
dr/d!
70.
Table P30.70 contains data taken for a ferromagnetic
material. (a) Construct a magnetization curve from the
data. Remember that
B " B
0
*
#
0
M. (b) Determine the
ratio B/B
0
for each pair of values of B and B
0
, and
construct a graph of B/B
0
versus B
0
. (The fraction B/B
0
is
called the relative permeability, and it is a measure of the
induced magnetic field.)
71.
A sphere of radius R has a uniform volume charge density
:
. Determine the magnetic field at the center of the
sphere when it rotates as a rigid object with angular speed
4
about an axis through its center (Fig. P30.71).
Figure P30.71 Problems 71 and 72.
R
4