MAGNETISM
Final Exam
DUE: Monday 2/22/10 - 5:00 pm
Tuesday 2/16/10
Name
DL119: period 4
JFW
CM:
(1)
(25)
(2)
(25)
(3)
(25)
(4)
(25)
(5)
(25)
(6)
(25)
TOTAL
(150)
A sphere of uniform charge distribution r and radius R spins with
constant angular velocity w about a diametric axis z. Derive the
equation for the magnetic moment m of the sphere.
Given the spinning sphere in the previous problem, calculate the total
magnetic energy stored in all space using Eq. 7.31, page 318. The expressions
for Ain and J are:
Ain
=
mowrrsinq
2
æ è
R2
3
-
r2
5
ö ø
^
f
J
=
wrrsinq
^
f
A sphere of radius a centered at the origin is made of an l.i.h.
conducting material of conductivity s. The electric potential on
the surface is maintained at the value, given in sphereical coordinates,
Vocosq, where Vo=constant.
(a) Using the techniques of chapter 3, show that the electric potential
inside the sphere is given by
V(r,q)
=
Vorcosq
a
(b) Using Ohm's law in the form
Jf = sE
determine the free current density Jf everywhere inside the sphere.
An electromagnetic wave is x-polarized and is traveling in the
direction from the origin to the point P(0,2,3). Assume that the fields are
real and the wave is traveling in a vacuum.
(a) Determine [^(n)], [^(k)], k·r, E, and B.
(b) Show that the fields are perpendicular to each other.
Consider a long, ideal solenoid of n turns per unit length, a
cross sectional area S, and a free current I in its windings. Assume
that the interior is filled completely with an l.i.h. magnetic material of
relative permeability, Km.
(a) Determine H inside and outside the solenoid.
(b) Determine B inside and outside the solenoid.
(c) Determine M inside the solenoid.
(d) Determine the self inductance L of a length l of the solenoid
and express it in terms of Lo, the self inductance of the solenoid when
it contains a vacuum.
The vector potential inside the outer metal conductor (b £ s £ c)
of a long cylindrical transmission line is:
A
=
ì í
î
Ao-
moI
4p
é ë
1+2 ln
æ è
b
a
ö ø
ù û
+
moI
4p(c2 -b2)
é ë
s2-b2-2c2ln
æ è
s
b
ö ø
ù û
ü ý
þ
^
z
(a) Determine the magnetic field B in the region
(b £ s £ c).
(b) Determine the expression for Ao for the case where
A=0 in the region s > c.
(c) If you did everything right, eliminating Ao gives
A
=
ì í
î
moIc2
4p(c2-b2)
é ë
s2
c2
-ln
æ è
s
c
ö ø
2
-1
ù û
ü ý
þ
^
z
(b £ s £ c)
To see how the vector potential and magnetic field vary in the outer
conductor, plot the following reduced functions in the range
0.9 £ u £ 1.
Ared
=
u2-ln u2-1
Bred
=
2(1-u2)
This problem addresses both poor and good conductors.
(a) Some free charge, highly localized, is imbedded in a piece of glass
and allowed to spread out. How long must one wait for the charge to decay
to 36.8% of its original value?
(b) Show that the skin depth in a good conductor
(s >> we) is l/2p, where l is the
wavelength inside the conductor. Find the numerical
value of the skin depth for the case
s = 107/W·m
w = 1015 rad/s
e » eo
m » mo
Why are metals opaque?
(c) Show that in a good metal the magnetic field lags the electric
field by 45o and find the ratio of the field amplitudes (Bo/Eo).
THIS PROBLEM IS FOR EXTRA CREDIT - FIVE EXTRA POINTS ON YOUR TOTAL
GRADE. It is not required that you do this problem.
A lens is coated with a thin film to affect the transmittance (T) and
reflectance (R) of the system. Consider a beam of light impinging at normal
incidence. Use the following values to determine R and T.
nair=1.00
nfilm=1.30
nlens=1.50
l = 680 nm
d=100 nm
Assume that the lens area struck by the beam is essentially flat so that you
can use the model presented in class. USE MAPLE.
We want to obtain the highest value of T using interference effects. Is the
thickness chosen, the best value (for cost sake, choose the minimum
thickness)?
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