Physics 200C
Dr. Cebra
Final Exam
10-June-09
Name: _________________
Instructions: Work all parts of each problem (each problem worth 50 points). No notes,
calculators, etc. Please write your name on every page – they will be separated. Staple
additional sheet to their respective problems.
1. Consider a light wave with p-polarization incident on a polished aluminum surface.
The real part of the index of refraction is n, while the imaginary part is k.
a) Determine the reflection coefficient, Rp, as a function of incident angle.
b) Draw a plot of Rp as a function of incident angle i.
c) Find the value of incident angle i for which Rp is a minimum.
Phys 200C
Name:_________________
 
2. For a plane wave, a0 eik0 r ' , incident on an aperture with the aperture function f0(r),
where f0 = 1 within the aperture and f0 = 0 otherwise, the diffracted amplitude in the
Fraunhofer approximation is


e ikr
a D 0 (r )  ia 0 k  nˆ '
2r

 

f 0 (r ' )e ik r ' dS '
S
Where k = k – k0 and k = kr
a) Show that if the screen includes two identical apertures separated by r, then the
diffracted intensity is

1 
a D a D*  4a D 0 a D* 0 cos 2 ( k  r )
2
b) Consider the case when each aperture is circular with a diameter d, the separation
between the aperture is r = 3d, and the wavelength on the incident light is  =
9d. Sketch the pattern that would be observed on a distant screen.
2
e
Remember:
d  2J 0 ( )
0
a
J
0
k0
i cos
0
(qr )rdr 
a
J 1 (qa )
q
Phys 200C
Name: _________________
3. Radiation from moving electrons:
a) The classical model of a hydrogen-like atom has a single electron in rotating in a
circular orbit of radius r around a point-like nucleus of charge Ze. Calculate the
fractional energy radiated per revolution, PT/Ek, where P is the radiated power, T is the
orbital period, and Ek is the kinetic energy of the electron.
b) The Large Electron-Positron Collider (LEP) accelerated electrons to an energy E >>
me and maintained them in circular orbits or radius R. Determine the fractional energy
radiated per revolution at LEP.
Phys 200C
Name: _________________
4. The scalar and vector potentials of an oscillating dipole with dipole moment p0 and
frequency  are respectively:
p  cos 
(r , , t )   0
sin t  r / c
40 c r

 p
A(r , , t )   0 0 sin t  r / ckˆ
4r
a) Show that these potentials are in the Lorentz gauge.
b) Find the electric and magnetic fields and the Poynting vector.
c) Find the total power radiated.
Phys 200C
Name: _________________
5. A magnetic monopole with magnetic charge qm passes through matter and losses
energy by collisions with electrons, just as does a particle with electric charge zqe. For
the charged particle the determination of the energy loss includes the following steps:
 
zq e E  v
dW

ds
v
2 2
z q 
k 02 v 2
 1 
dW
  2 e 2  d Im 
ln(
1

)

ds
4 v o
2
  ( ) 
z 2 q e2
dW
k v 

ln  o   d Im  
2 2 2
ds
2  o v
 0
 mv 2
ne z 2 q e4
dW


ln
ds
4o2 mv 2  



a) Show that the energy loss per unit distance for a monopole is given approximately
by:
 2 2 mv 2 
q m2 qe2
 dW 

 ne
ln 


402 mc 2   
 ds  monopole
b) Sketch the dW/ds curve for a monopole and for a charged particle of similar mass.
Discuss the similarities and the differences.
 cq q
n
c) With the Dirac quantization condition 0 e m  determining the magnetic
4
2
charge, what z value is necessary for an ordinary charged particle in order that it
lose energy at relativistic speeds at the same rate as a monopole? [Hint, recall that
the fine structure constant   qe2 / 4o c ]