Massachusetts Institute of Technology
Department of Physics
Physics 8.04 Thu Oct 12 16:37:47 EDT 1995
As in the syllabus, ``FT''= French and Taylor, ``ER''= Eisberg
and Resnick, ``FLP''= Feynman Lectures, vol. III, and the
numbers refer to sections. Here the numbers listed are the
pre-lecture readings. The ``extended'' readings are also given, but
within parenthesis. Reading assignments:
Lecture 1, Tue. Feb. 7: FT 1.1-1.4.
Lecture 2, Thurs. Feb. 9: ER 1.1-1.4 (ER 1.5-1.7).
Lecture 3, Tue. Feb. 14: FT 1.6 (7 pages) (ER 2.1-2.6).
Lecture 4, Thurs. Feb. 16: FT 2.1-2.2, 2.4 (FT 2.3, 2.5-2.10).
(Due Friday, February 17 at 5:00 pm sharp.)
Work requested of the students is highlighted by either a question mark or underlining.
1. (3 pts) Computer assignment # 1
Later problem sets in the course may involve running programs on Athena. To make sure everyone in the class is familiar with Athena and to open another channel of communication between the teaching staff and the class, send email to muchomas@mit.edu with the subject heading ``8.04s''. In the body of the letter please send one line containing an estimate of the number of hours you spent working on this problem set.
2. (5 pts) Practice with Units
In this class we shall use mostly Gaussian CGS units, which are those usually used in 8.022 and the the text, Electricity and Magnetism, vol. 2 of the Berkeley Physics Course. In these units Maxwell's equations are
where 
is the electric field, 
is the magnetic field,
is the current density, c is the speed of light. The
Lorentz force 
on a particle of charge q traveling at
velocity 
in an electric field 
and magnetic field
is then written
a) From (1) derive the form 
for the
electric arising from a point charge 
at the origin. From your
result and (5) give the force 
between two particles
of charge 
and 
as a function of the distance between them,
r. Finally, from 
give the potential energy U
between two charges 
and 
separated by a distance r.
b) In CGS units, the unit of energy in measuring U is the erg (
-
) and the unit of distance in measuring r is the centimeter. For the units to agree on both sides of your expression for
U, what then, in terms of g, cm and s (grams, centimeters and
seconds) must be the units associated with charge in this system? This
combination of units is conventionally termed the esu, ``the
electrostatic unit'' and is the basic unit of charge in the Gaussian CGS
system. The electrostatic unit is defined so that the force between two
point charges, each carrying a charge of 1 esu, at a distance of 1 cm
is 
-
c) Using your result in a), in terms of the esu and the cm what
units does 
carry in this system? For the units to balance
correctly in (5), what relationship must exist between the units
carried by 
and 
? Now, verify explicitly
that the units in (1)-(5) balance.
d) Using the values 
, 
, 
, 
-
compute the values (and units) of
(``Bohr radius'')
(``Hartree'')
(``Fine structure constant'')
e) Give the three quantities in d) in terms of
Angstroms (1 Å 
) and electron-Volts (
-cm).
3. (7 pts) J.J. Thomson's measurement of e/m
Do problem 1.4 from French and Taylor (p. 45).
Also, in terms of E, B, l, and 
and c, what deflection
angle should be expected if, instead, the magnetic field were
maintained but the electric field shut off? You may assume that the
fields are sufficiently weak that the electrons do not collide with
the plates of the apparatus. (Hint: first find the radius of
curvature of the electrons' trajectory in the magnetic field.)
4. (7 pts) J.J. Thomson's Model of the Atom
(You may wish to refer to problem 1.6 in French and Taylor.)
a) Starting with 
,
derive the form for the electric field 
throughout all of space in the presence of the positive charge cloud
of a Thomson Model atom of charge Ze and radius R. (Take the
positive charge distribution to be of a uniform density).
Determine also the electrostatic potential 
for
all r.
b) From Newton's law 
, write down
the equation of motion for the coordinate 
of a single
electron of charge 
inside of such an atom in terms of
, 
, the charge and mass of the
electron and the parameters describing the atom. Separate the vector
equation into three non-coupled equations for each of the directions
, 
, 
.
c) From the equations derived in b), write down the
most general solution for the motion of the electron, 
, for an orbit contained entirely within the atom. Is this
motion periodic (does the electron return to the same point again and
again)? What is the period of the motion and what frequency of
electromagnetic radiation would you expect such an electron to emit?
5. (10 pts) Detailed analysis of scattering: Differential Scattering Cross Section
In this problem you will explore the detailed analysis of scattering
using the simple model of point particles scattering from
infinitely massive hard spheres.
Consider the single collision event in the diagram in Figure
(the x-axis runs parallel to the incoming trajectory and
directly through the center of the sphere):
a) In a frictionless hard-wall collision with a
sphere so massive so that its recoil may be ignored,
the angle of incidence of the particle to the sphere's
surface will equal the angle of departure from the sphere's surface.
Under these assumptions, compute the scattering angle
for a particle hitting the sphere off-center, a distance
b from the x-axis. (b is called the ``impact parameter'').
Describe what will happen if the particle still hits the
sphere a distance b off-axis but hits the sphere at a point out of
the plane of the diagram.
b) Imagine that a steady and uniform flux density of J
particles per unit area per unit time directed along 
is
incident on the sphere. How many particles will impact the sphere per
unit time with impact parameters in the range from 
to 
?
(Keep terms to first order in db only.)
c) Into what range of angles 
will the sphere
scatter the particles with the impact parameter range considered in
b)? Do you expect the particles to be scattered evenly among
these angles? What solid angle does this range of angles subtend?
d) How many particles per unit solid angle per unit time, P, are
scattered in the direction 
? Express
your result in the form
The quantity you have calculated, 
, is known as
the ``differential scattering cross section.'' Note that it takes a
particularly simple form when measuring scattering from spheres.
e) Now imagine a thin foil (thickness t) of a material consisting of
these spheres (still radius R) with a density of n spheres per unit
volume. Normally incident on this foil we direct a current of I
particles per unit time spread over a wide region of the foil. (See
Figure .)
Assume for simplicity that the foil is sufficiently thin and the spheres
sufficiently small compared to their average spacing that that the
spheres do not occult one another and that the vast majority of particles
pass through the foil colliding with at most one sphere. What
fraction of the particles can be expected to pass through the foil
without a single collision? Again ignoring multiple scattering events,
at what rate (particles/unit time) will particles emerge from the foil
with deflection angles between 30
and 90
away from the 
direction of incidence?
6. (10 pts) Counting Normal Modes
a) Imagine a string constrained to move in the 
direction only, obeying the wave equation
where c is the wave velocity along the string. (See Figure
.) Because the string is tied to the wall at both ends, we
also have the boundary conditions that
.
If we seek solutions
of the form 
, we will find that 
must obey
with boundary conditions 
. Write down the most
general form of the solution to (7) which is consistent
with these boundary conditions. For each solution, give also
the corresponding values of the angular frequency 
. These are
the normal modes. (We do not consider the solution 
to be a
mode.)
b) Define a function 
which tells how many modes
have angular frequency less than 
. Keeping
in mind that 
is a discontinuous function,
sketch a plot of 
.
c) In terms of 
, how many modes per unit
length have frequencies in the range from 
to
(including 
but not
)? If one were to approximate this quantity
by 
, what would
be the maximum discrepancy between this approximation and the actual
value? Under what circumstances, then, would you regard 
as a good approximation?
d) Now imagine the same string (with the same wave velocity
c and length L) wrapped around a cylinder of radius 
.
(See Figure .) Equation (7) still holds but now
(because the string is tied to itself) the boundary conditions are
and 
. Determine the normal
modes and frequencies under these circumstances and
sketch this new 
on your previous sketch of
. How do 
and 
compare?
Discuss the effect of changing the boundary conditions on
the number of modes per unit length per unit frequency interval.
7. (10 pts) Mode counting in two dimensions
Repeat the mode counting analysis given in lecture but
for 
instead of 
and proceeding in two dimensions
instead of three. In this example consider a rubber sheet stretched
over a square frame of dimensions 
. (See Figure .)
You may take the speed of sound of the sheet to be c, and you need
consider only the single polarization of up-and-down motions of the
sheet.
Show that the number of modes per unit area per unit
frequency interval is independent of the aspect ratio 
of the
sheet.
