Download Physics 322 Midterm 2 1 (15 pt) 2 (50 pt) 3 (20 pt) 4 (15 pt) total (100

Physics 322 Midterm 2
Nov 30, 2015
name:
Box your final answer.
1 (15 pt)
2 (50 pt)
3 (20 pt)
4 (15 pt)
total (100 pt)
1
~ = −M ẑ. Compute
1. (15 pt) An infinitely long cylinder of radius R whose axis is parallel to the ẑ axis has a uniform magnetization M
~
H everywhere.
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2. Suppose for t < 0, the capacitor is charged to V0 in the circuit below.
The switch is closed at t = 0.
a) (10 pt) The current behaves as a function of time as I(t) = (envelope function of time) × cos ( f t + δ ) where f > 0 for (R/L)2 <
4/(LC). What is f ?
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b) (15 pt) Suppose the inductance L is due to the self-inductance of a toroidal coil with rectangular cross section (innder radius a, outer
radius b, and height h as shown in the figure below) carrying a total of N turns. Express L in terms of N, h, a, and b.
c) (10 pt) Suppose the toroidal coil described in part b) has a magnetic field ~B(s, φ , z) = f (s)φ̂ (where s is a cylindrical coordinate
variable: e.g. the inner radius is described as s = a). In this magnetic field, suppose a pointlike magnetic dipole with ~m = mẑ is
placed at rest at s = u ∈ (a, b) inside the torus. What are the forces and torques on the dipole due to the magnetic field?
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d) (15 pt) Suppose the capacitor is made of parallel circular plates each with radius R with vacuum in between. Suppose the charge
density on one of the plates for t > 0 is σ (t) and the uniform electric field region between the capaciator plates is then characterized
by
~E ≈ σ (t) ẑ.
ε0
Neglecting the edge effects as usual and assuming rotational symmetry about the axis passing through the center of the plates, find
the resulting Bφ and the Poynting vector at s = R, somewhere between the parallel plates in the uniform field region.
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3. Suppose you are given that the Maxwell equations were replaced by
~∇ · ~Ẽ = 0
~
~
~∇ × ~Ẽ = − ∂ B̃ − L̃ ∂ Ẽ
∂t
∂t
~
~∇ ×~B̃ = Ñ ∂ Ẽ − L̃~∇ × ~Ẽ
∂t
~∇ ·~B̃ = 0
where L̃ and Ñ are complex constants and the tilde indicates that these fields are complex.
a) (10 pt) What is the wave equation governing ~Ẽ?
b) (10 pt) Find a plane wave solution ~Ẽ to this Maxwell equation system proportional to e−iωt , traveling in the y direction, having a
single wavelength, and the boundary condition
~Ẽ(t = 0,~x = 0) = ~C̃.
Be sure to specify the condition on ~C̃ coming from the Maxwell equation system.
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4. (15 pt) A plane wave is incident normal to a dielectric interface as shown:
where the index of refraction of medium i ∈ {1, 2} is ni and µ1 = µ2 . The incident wave is
~Ẽ (t, z) = Ẽ exp [i (k z − ωt)] x̂
I
0I
1
and the electric field in the rest of the regions can be written as
~Ẽ (t, z) = Ẽ exp [i (−k z − ωt)] x̂
R
0R
1
~Ẽ (t, z) = Ẽ exp [i (k z − ωt)] x̂.
T
0T
2
Recall that one can derive from one of the Maxwell equations the magnetic field for the reflected wave being
ω µ1 ε1
B̃~R (t, z) = −
Ẽ0R ei(−k1 z−ωt) ŷ
k1
~ k continuity
where the minus sign in the amplitude comes from the minus sign in~k = −k1 ẑ for the reflected wave. Use ~Ek continuity and H
to find ~Ẽ just in terms of {~Ẽ , n , n }.
0T
0I
1
2
7
(4 continued) (extra space )
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