| The Vector Potential and the Vector Poisson Equation |
8.1.1 | A solenoid has radius a, length d, and turns N, as shown in Fig.
8.2.3. The length d is much greater than a, so it can be
regarded as being infinite. It is driven by a current i.
(a) | Show that Ampère's differential law and the magnetic flux
continuity law [(8.0.1) and (8.0.2)], as well as the associated
continuity conditions [(8.0.3) and (8.0.4)], are satisfied by an
interior magnetic field intensity that is uniform and an exterior one
that is zero.
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(b) | What is the interior field?
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(c) | A is continuous at r = a because otherwise the
H field would have a singularity. Determine A.
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8.1.2* | A two-dimensional magnetic quadrupole is composed of four line
currents of magnitudes i, two in the positive z direction at x = 0,
y = d/2 and two in the negative z direction at x = d/2, y
= 0. (With the line charges representing line currents, the
cross-section is the same as shown in Fig. P4.4.3.)
Show that in the limit where r d, Az = - ( o
id2/4 )(r-2) cos 2 . (Note that distances must be
approximated accurately to order d2.)
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8.1.3 | A two-dimensional coil, shown in cross-section in Fig. P8.1.3, is
composed of N turns of length l in the z direction that is much
greater than the width w or spacing d. The thickness of the windings
in the y direction is much less than w and d. Each turn carries the
current i. Determine A.
Figure P8.1.3
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| The Biot-Savart Superposition Integral |
8.2.1* | The washer-shaped coil shown in Fig. P8.2.1 has a thickness
that is much less than the inner radius b and outer
radius a. It supports a current density J = Jo i .
Show that along the z axis,
Figure P8.2.1
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8.2.2* | A coil is wound so that the wire forms a spherical shell of
radius R with the wire essentially running in the direction.
With the wire driven by a current source, the resulting current
distribution is a surface current at r = R having the density
K = Ko sin i , where Ko is a given constant.
There are no other currents. Show that at the center of the
coil, H = (2Ko/3)iz.
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8.2.3 | In the configuration of Prob. 8.2.2, the surface current density
is uniformly distributed, so that K = Ko i , where
Ko is again a constant. Find H at the center of the coil.
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8.2.4 | Within a spherical region of radius R, the current density is
J = Jo i , where Jo is a given constant. Outside
this region is free space and no other sources of H. Determine
H at the origin.
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8.2.5* | A current i circulates around a loop having the shape of an
equilateral triangle having sides of length d, as shown in Fig.
P8.2.5. The loop is in the z = 0 plane. Show that along the z
axis,
Figure P8.2.5
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8.2.6 | For the two-dimensional coil of Prob. 8.1.3, use the Biot-Savart
superposition integral to find H along the x axis.
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8.2.7* | Show that A induced at point P by the current stick of Figs.
8.2.5 and 8.2.6 is
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| The Scalar Magnetic Potential |
8.3.1 | Evaluate the H field on the axis of a circular loop of radius R
carrying a current i. Show that your result is consistent with the
result of Example 8.3.2 at distances from the loop much greater than
R.
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8.3.2 | Determine for two infinitely long parallel thin wires carrying
currents i in opposite directions parallel to the z axis of a
Cartesian coordinate system and located along x = a. Show that
the lines = const in the x - y plane are circles.
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8.3.3 | Find the scalar potential on the axis of a stack of circular
loops (a coil) of N turns and length l using 8.3.12 for an
individual turn, integrating over all the turns. Find H on the
axis.
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| Magnetoquasistatic Fields in the Presence of Perfect Conductors
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8.4.1* | A current loop of radius R is at the center of a conducting
spherical shell having radius b. Assume that R b and
that i(t) is so rapidly varying that the shell can be taken as
perfectly conducting. Show that in spherical coordinates, where R
r < b
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8.4.2 | The two-dimensional magnetic dipole of Example 8.1.2 is at the
center of a conducting shell having radius a d. The current
i(t) is so rapidly varying that the shell can be regarded as
perfectly conducting. What are and H in the region d
r < a?
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8.4.3* | The cross-section of a two-dimensional system is shown in Fig.
P8.4.3. A magnetic flux per unit length s o Ho is trapped
between perfectly conducting plane parallel plates that extend to
infinity to the left and right. At the origin on the lower plate is a
perfectly conducting half-cylinder of radius R.
(a) | Show that if s R, then
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(b) | Show that a plot of H would appear as in the left half
of Fig. 8.4.2 turned on its side.
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Figure P8.4.3
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8.4.4 | In a three-dimensional version of that shown in Fig. P8.4.3, a
perfectly conducting hemispherical bump of radius s R is attached
to the lower of two perfectly conducting plane parallel plates. The
hemisphere is centered at the origin of a spherical coordinate system
such as in Fig. P8.4.3, with . The magnetic field
intensity is uniform far from the hemisphere. Determine and
H.
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8.4.5* | Running from z = - to z = + at (x, y) = (0, -h) is
a wire. The wire is parallel to a perfectly conducting plane at y = 0.
When t = 0, a current step i = I u-1(t) is applied in the +z
direction to the wire.
(a) | Show that in the region y < 0,
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(b) | Show that the surface current density at y = 0 is
Kz = - ih/ (x2 + h2).
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Figure P8.4.6
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8.4.6 | The cross-section of a system that extends to infinity in the
z directions is shown in Fig. P8.4.6. Surrounded by free space,
a sheet of current has the surface current density Ko iz
uniformly distributed between x = b and x = a. The plane x = 0
is perfectly conducting.
(a) | Determine in the region 0 < x.
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(b) | Find K in the plane x = 0.
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| Piece-Wise Magnetic Fields |
8.5.1* | The cross-section of a cylindrical winding is shown in Fig.
P8.5.1. As projected onto the y = 0 plane, the number of turns per unit
length is constant and equal to N/2R. The cylinder can be modeled as
infinitely long in the axial direction.
Figure P8.5.1
Figure P8.5.2
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8.5.2 | The cross-section of a rotor, coaxial with a perfectly conducting
"magnetic shield," is shown in Fig. P8.5.2. Windings consisting of N
turns per unit peripheral length are distributed uniformly at r = b so
that at a given instant in time, the surface current distribution is as
shown. At r = a, there is the inner surface of a perfect conductor. The
system is very long in the z direction.
(a) | What are the continuity
conditions on at r = b and the boundary condition at r = a?
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(b) | Find , and hence H, in regions (a) and (b) outside
and inside the winding, respectively.
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(c) | With the understanding that the rotor is
wound using one wire, so that each turn is in series with the next and
a wire carrying the current in the +z direction at returns the
current in the -z direction at - , what is the inductance of the
rotor coil? Why is it independent of the rotor position o?
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| Vector Potential |
8.6.1* | In Example 1.4.1, the magnetic field intensity is determined to
be that given by (1.4.7). Define Az to be zero at the origin.
(a) | Show that if H is to be finite in the neighborhood
of r = R, Az must be continuous there.
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(b) | Show that A is given by
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(c) | The loop designated by C' in Fig. 1.4.2 has a
length l in the z direction, an inner leg at r = 0,
and an outer leg at r = a > R. Use A
to show that the flux linked is
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8.6.2 | For the configuration of Prob. 1.4.2, define Az as being zero
at the origin.
(a) | Determine Az in the regions r < b and b < r < a.
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(b) |
Use A to determine the flux linked by a closed rectangular loop
having length \l in the z direction and each of its four sides in a
plane of constant . Two of the sides are parallel to the z axis,
one at radius r = c and the other at r = 0. The other two, respectively,
join the ends of these segments, running radially from r = 0 to r = c.
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8.6.3* | In cylindrical coordinates, o H = o [Hr(r, z)ir + Hz(r,
z)iz].
That is, the magnetic flux density is axially symmetric and does not
have a component.
(a) | Show that
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(b) | Show that the flux passing between contours at r = a and
r = b is
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8.6.4* | For the inductive attenuator considered in Example 8.6.3 and
Demonstration 8.6.2:
(a) | derive the vector potential, (20), without identifying this MQS problem with its EQS counterpart.
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(b) | Show that the current is as given by (21).
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(c) | In the limit where b/a 1,
show that the response has the dependence on b/a shown
in the plot of Fig. 8.6.11.
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(d) | Show that in the opposite limit, where
b/a 1, the total current in the lower plate (21)
is consistent with a magnetic field intensity between the upper and
lower plates that is uniform (with respect to y) and hence equal to
( /b o )ix. Note that
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Figure P8.6.5
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8.6.5 | Perfectly conducting electrodes are composed of sheets bent into
the shape of 's, as shown in Fig. P8.6.5. The length of the
system in the z direction is very large compared to the length 2a or
height d, so the fields can be regarded as two dimensional. The
insulating gaps have a width that is small compared to all
dimensions. Passing through these gaps is a magnetic flux (per unit
length in the z direction) (t). One method of solution
is suggested by Example 6.6.3.
(a) | Find A in regions (a) and (b) to the right and left, respectively, of the plane x = 0.
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(b) | Sketch H.
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8.6.6* | The wires comprising the winding shown in cross-section by Fig.
P8.6.6 carry current in the -z direction over the range 0 < x < a and
return this current over the range -a < x < 0. These windings extend
uniformly over the range 0 < y < b. Thus, the current density in the
region of interest is J = - ino sin ( x/a)iz, where i
is the current carried by each wire and |no sin ( x/a)| is the
number of turns per unit area. This region is surrounded by perfectly
conducting walls at y = 0 and y = b and at x = -a and x = a.
The length l in the z direction is much greater than either a or b.
Figure P8.6.6
(a) | Show that
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(b) | Show that the inductance of the winding is
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(c) | Sketch H.
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8.6.7 | In the configuration of Prob. 8.6.6, the rectangular region is
uniformly filled with wires that all carry their current in the z
direction. There are no of these wires per unit area. The
current carried by each wire is returned in the perfectly conducting
walls.
(a) | Determine A.
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(b) | Assume that all the wires are connected to the wall by a
terminating plate at z = l and that each is driven by a current
source i(t) in the plane z = 0. Note that it has been assumed
that each of these current sources is the same function of time. What
is the voltage v(x, y, t) of these sources?
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8.6.8 | In the configuration of Prob. 8.6.6, the turns are uniformly
distributed. Thus, no is a constant representing the number of
wires per unit area carrying current in the -z direction in the region
0 < x. Assume that the wire carrying current in the -z direction at the
location (x, y) returns the current at (-x, y).
(a) | Determine A.
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(b) | Find the inductance L.
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