Unformatted text preview: PHY 3323 December 5, 2011
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(1) Consider two loops of radius R which carry current I counterclockwise. Each loop is
parallel to the x-y plane, and is centered on the z axis. The top loop is at a height d/2
above the x-y plane, and the bottom loop is d/2 below the x-y plane.
a) Use the Biot-Savart Law to ﬁnd the total magnetic ﬁeld vector at position z on
the z axis. (30 points)
b) Suppose there is a value of the separation d such that ∂ 2 B/∂z 2 vanishes at z = 0.
On the basis of dimensional analysis, what must be the forms of d and the value
of B at z = 0 for this value of d? (30 points)
c) Find d and the value of B at z = 0 for this value of d. (20 points)
(2) A uniformly charged solid sphere of radius R carries a total charge Q, and is set
spinning with angular velocity ω about the z axis.
a) What form must the magnetic dipole moment m of the sphere take on the basis of
dimensional analysis? (20 points)
b) What is the current density inside the sphere? (20 points)
c) What is the magnetic dipole moment of the sphere? (20 points)
d) What is the leading large distance form of the vector potential? (20 points)
(3) At the interface (the x-y plane) between one linear magnetic material and another the
magentic ﬁeld lines bend as follows:
z>0 =⇒ µ1 and B1 = −B1 cos(θ1 )z + sin(θ1 )x , z<0 =⇒ µ2 and B2 = −B2 cos(θ2 )z + sin(θ2 )x . a) What is the ﬁeld H in each region? (30 points)
b) Use the boundary conditions at the interface to solve for B2 and θ2 in terms of B1
and θ1 . (30 points)
c) What is the magnetization density M in each region? (30 points). ...
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