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(1) [18 points I Two long cylindrical conducting shells with radius a and b are arranged
concentrically, with a < b, and are charged so that their potentials are Va and Vb, respectively.
(a) Write down the general form of the electric field between the shells.
(b) Find the potential at points between the shells.
(c) Now supposing that Vb = 0, but that V remains finite, find the energy stored in a length L of
this system, in terms of Va and the system dimensions L, a, and b.
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the electric potential, V k F with k
(2) 110 points I (a) Find the electric field corresponding to
that could produce such a field.
a fixed vector, and give a physical example of a situation
ine whether it can exist in electrostatics,
(b) For each of the following two electric fields, determ
e it. a and /3 are constants.
and if it can exist, determine a charge density that could produc
(i) E=a(x-y); (ii) E=/3(x+y’)
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the vector potential A = —bIn(s)
(3) 110 pointsj (a) Find the magnetic field corresponding to
le of a physical situation that would
(in cylindrical coordinates; b is constant) and give an examp
produce this field.
r it can exist in magnetostatics,
(b) For each of the following magnetic fields, determine whethe
e it. y and ij are constants.
and if it can exist, determine a current density that could produc
(i) B=y(x’—y); (ii) B=ij(xi+)
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(4) [17 pointsl A long grounded conducting cylindrical rod with radius a is placed in a uniform
, directed in the x direction, perpendicular to the axis of the
0
electric field, of magnitude E
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cyhnder.
cylinder.
the
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(a) Find the electric field
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(b) Find the charge density vs. angle on the surface of the ro,
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d about the origin, has a uniform
(5) [18 pointsj A sphere of ferromagnetic material, centere
2 with M) a constant. The sphere has radius R.
0
magnetization, M
(a) Find the bound currents.
(b) Find the total dipole moment of this sphere.
M
0
= p
(2cosO + sin oo) ,find the
(c) Given that the magnetic field outside the sphere is B
this field is derived by using the
magnetic field at points just inside the sphere, showing how
of
9 and (and note that
appropriate boundary conditions. Give the final solution in terms
this shquld simplify the answer).
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(6) 117 pointsj A parallel-plate capacitor has area A and plate separation d. The lower plate
carries free charge +Q, and sits on the x-’’ plane at z = 0. The upper plate sits at z = d and has free
charge —Q. The interior of the capacitor is filled with a dielectric whose relative permittivity
changes with z according to, £r e’ with a being a constant.
(a) Find the displacement field, D, inside the capacitor.
(b) Find the electric field in the capacitor, and the potential difference between the plates.
(c) Find the surface and volume bound charge densities in the dielectric.
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density,
(7)110 points I A long wire, radius R, carries a current along its length with current
.
2
i = as
(a) Find the total current carried by the wire.
(b) Using Ampere’s law find the magnetic field, with direction, both for points s < R and
s>R.
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