EXAM 3                December 3, 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.     Magnetic transducers are often used to monitor small
vibrations.  For example, the end of a vibrating bar may be
attached to an N-loop coil which in turn vibrates in and out of a uniform
magnetic field  B, as shown in the                    ___________
figure to the right.  Show that                      |X X X      |     <----
the speed of the end of the bar,                     |X X X      |
dx/dt, is related to the EMF                         |___________|     ----->
induced in the coil by
   
     E = N B b dx/dt .

2.     A straight wire of radius  a  placed along the axis of
a cylindrical metal shell with inner radius  b  and outer radius c , 
carries a current  I.  The shell carries a current
I  in a direction opposite to that in the wire.  Find the magnetic
field  B  for  a < r < b  and also for r > c .  Coaxial cables of this
type are used widely to minimize unwanted magnetic effects.

3. (From homework)  A resistor  R and an inductor L are

connected in parallel across an AC generator. The source

EMF is  E = E  sin (omega t).  Determine
             0
the current through the resistor and the inductor (including

amplitude and phase) as a function of time.  What is the total

current leaving the source (you need not make I    algebraically
                                               tot
pretty; any correct expression is acceptable).


4.     So-called Helmholtz coils are sometimes used to obtain a
uniform magnetic field in
situations where a solenoid would be           <----L---->
impractical.  Two coils of radius a, 
placed as shown in the figure to the           O ---P--- O
right, are used. Show that, if                           ^ coil of radius a 
L = a, then B at the point P has the value                 perpendicular to line

             2
      mu  I a
        0            2    2  -3/2       2          2  -3/2
  B = ---------  [ (a  + x  )      + (2a  - 2ax + x  )     ]
          2