EXAM 1                October 7, 1987               PS 208    
  
       
  
     If you need a constant not given here or do not understand a 
  
problem, please ask me about it.  No notes, books, etc. may be 
  
used during this exam.   
  
  
               2                             -19
  g = 9.8 m/sec                  e = 1.6 x 10    C   
  
  
                               9     2    2
  1/(4 pi epsilon )  = 9.0 x 10   N m  / C
                 0
                                     -31
The mass of the electron is 9.11 x 10    kg.   
  
  
  
  
  
1.     Two point charges  Q   and  Q    are a distance  d  apart, 
                           1        2
  
and their combined charge is  Q    .  If they repel each other 
                               tot
  
with a force  F,  find the two charges in terms of  Q  , d , and 
                                                     tot
  
F.  
  
  
  
  
  
2.     A force  F = + kx  acts on a particle of mass  m.  
  
  
  (a)  Show that the differential equation governing the motion of 
  
                      2
                     d x
the particle is  m   ---   - kx = 0 .    
                       2
                     dt
  
                  alpha t
  (b)  Show that e        is a solution of this equation and find the 
  
appropriate value or values of  alpha .  
  
  
  (c)  Interpret the solution physically (i. e. by describing it 
  
qualitatively or by sketching it).  What is wrong with this 
  
motion, and where does the problem arise?  
  
  
  
  
  
3. (From homework).     A thin cylindrical shell of radius  R   is 
                                                             1
  
surrounded by a second concentric cylindrical shell of radius  
  
  
R  .  The inner shell has a total charge of  - Q   and the outer 
 2                                              1
  
shell  + Q , with  Q  > Q > 0 .  Assuming the length  L  of the shells 
          2         2    1
  
is much greater than either radius, determine the electric field 
  
  
as a function of  r (the perpendicular distance from the common 
  
  
axis of the cylinders) for all values of  r .
  
  
  
  
  
4.     A long thin rod is bent into the form of a semicircle of radius    

b , with its ends on the x axis at  x=+b  and  x=-b.  The rod carries 

a charge density  
  
                  lambda = A cos (theta), 

where  A  is a constant and the angle  theta  is measured from the right 

end. Find the y-component of the electric field at the origin (the center 

of the circle).