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on the larger remainder of the
spherical cap covering angles from 8= 0 to r14, and
spherical surface. Assume o is positive.
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(a) Find the dipole moment of this charge distribution, measured from the center of the sphere.
of
(b) Find the electric field, with direction, measured a long distance z directly above the center
the sphere, in the direction 0 = 0 (e.g. find the leading term at this point).
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(2) In some region of space, a steady magnetic field is given by, B = Cx
constant.
(a) Find the current density in this region.
ulation,
(b) Show that this current density satisfies the continuity equation with no charge accum
as expected for a steady current distribution.
(c) Find the vector potential associated with this current. [1 suggest using the gauge choice
discussed in class, though as always for such problems there are other solutions that may work as
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(3) A grounded metal sphere, radius R, is placed in the center of a linear
By an arrangement of external fields, a
radius 2R. The relative permittivity of the dielectric is
(at 2R), where a is a
potential V = acosO is generated at the outer surface of the dielectric
constant.
dielectric, R < r < 2R.
(a) Write the general form for the electric potential in the region of the
in terms of a and
(b) Using appropriate surface conditions, solve for the potential in this region
R.
(c) Find the electric field and the displacement field in this region.
and find the
(d) Find the free charge distribution on the grounded sphere at the center,
s.
distribution of bound charges in the dielectric and on the inner and outer surface
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(4) A hollow sphere has a single layer of windings around the z axis ( direction), with the turn
0 is a constant.
density (turns/length) equal to no sin8 where n
0 is large, the resulting current density can be approximated a
(a) If the wires are very thin and n
surface current. Find K as a function of 0 if the current in each wire is 1.
rsinO) inside the
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2 are constants. For these vector
1 and C
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sphere, and (c
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potentials, solve for the corresponding magnetic fields.
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(5) Consider a long wire, shown in cross-sect
circle is a hollow region (radius R!2) running
current density J into the page, while the smaller
the length of the wire.
inside and outside of a solid wire carrying
(a) First show how to calculate the magnetic field
were no hollow region.
uniform current, such as the shaded wire if there
ibution shown, at points along the horizontal
(b) Now find the magnetic field of the current distr
the hollow region. [Hint: think
diameter, both in the solid region of the wire and in
superposition.I
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