Assignment 1 (due 1/23/02)
Assignment 2 (due 1/25/02)
Assignment 3 (due 1/28/02)
Assignment 4 (due 1/30/02)
Assignment 5 (due 2/1/02)
Assignment 6 (due 2/4/02)
Assignment 7 (due 2/6/02)
Assignment 8 (due 2/8/02)
Assignment 9 (due 2/11/02)
Assignment 10 (due 2/13/02)
Assignment 11 (due 2/18/02)
Assignment 12 (due 2/20/02)
Assignment 13 (due 2/22/02)
Assignment 14 (due 2/25/02)
Assignment 15 (due 2/27/02)
Assignment 16 (due 3/1/02)
Assignment 17 (due 3/4/02)
Assignment 18 (due 3/6/02)
Assignment 19 (due 3/8/02)
Assignment 20 (due 3/11/02)
Assignment 21 (due 3/13/02)
Assignment 22 (due 4/3/02)
Assignment 23 (due 4/5/02)
Assignment 24 (due 4/8/02)
Assignment 25 (due 4/10/02)
Assignment 26 (due 4/12/02)
Assignment 27 (due 4/15/02)
Assignment 28 (due 4/17/02)
Assignment 29 (due 4/19/02)
Assignment 30 (due 4/24/02)
Assignment 31 (due 4/29/02)
Assignment 32 (due 5/1/02)
Assignment 33 (due 5/3/02)
Assignment 34 (due 5/6/02)
Assignment 35 (due 5/8/02)
Assignment 36 (due 5/10/02)
Assignment 1 (due Wednesday 1/23/02)
1. The CO2 molecule can be modeled as a central mass M2 connected by identical springs of spring constant k to two masses of mass M1.
O===C===O
(a) Set up and solve the equations for the two normal modes
of oscillation along the axis of the molecule. Use the matrix
technique you learned in class.
(b) Calculate the numerical ratio of the normal mode frequencies.
(c) Calculate the normal mode frequencies and identify the part
of the electromagnetic spectrum to which they correspond (e.g.
x-rays, visible light, radio waves, etc.). The CRC Handbook of
Chemistry and Physics lists the spring constant for a C=O double
bond as 16.00 N/cm, and each nucleon weighs 1.67 x 10-27 kg.
(d) How can these frequencies be measured experimentally?
(e) Describe (in words and sketches) the normal mode oscillations.
2. A torsion oscillator can be made by connecting a disk of moment of inertia I to a fixed point with a torsion rod. If the disk is twisted by an angle q from its equilibrium position, the torsion rod exerts a restoring torque t = -kq. Because the torque is proportional to the angular acceleration (t = I d2q/dt2), the system can undergo simple harmonic oscillations. For more information on torsion oscillators, see HRW pp. 379-80.
]===[]===[]===[
.kA..I..kC..I..kB..
(a) Calculate the normal mode frequencies of the oscillator, assuming
kA
and kB
are unequal and kC2 = kAkB
(b) Describe the normal modes of the system.
(c) Now suppose that kA
= kB
= kC
= k. If the system is released from
rest with the left-hand disk initially twisted from its equilibrium
position and the right-hand disk in its equilibrium position,
describe the resulting motion.
(d) How long does it take for the right-hand disk to get all the
kinetic energy. (Think of the motion as a linear combination of
normal modes.)
Reading: Today's material was covered in HL Ch 1 and 3, and you can review eigenvalue problems with matrices in your linear algebra text. For Wednesday, please read HL Ch 2. For a more basic introduction to mechanical waves, look at HRW Ch 17.
1. N identical point masses are evenly spaced along a massless string. Both ends of the string are attached to fixed points. As we have seen in class, the time-dependent motion of the jth mass in the nth mode is:
xn,j = C sin [npj/(N+1)] cos wnt,
where wn is the angular frequency of the nth normal mode. (To be thorough, we should also include an arbitrary phase shift in the argument of the cosine, but we'll omit it here for simplicity.) Your goal is to prove that there are exactly N unique normal modes. Consider both amplitude and frequency in your proof.
(a) Show that in mode numbers n=0 and n=N+1, all the oscillators
have zero amplitude.
(b) Show that mode numbers +n and -n correspond to the same oscillation.
(c) Show that mode numbers (N+1+n) and (N+1-n) correspond to the
same oscillation.
(d) Based on a, b, and c, argue that there are exactly N unique
normal modes.
2. The crystal lattice of solids can be modeled as a three-dimensional network of point masses (atoms) connected by springs (bonds). To simplify matters, let's consider a salt crystal NaCl, thinking of it as just a one-dimensional chain of masses and springs. The analog of the natural frequency w0 = p/l (Y/r)1/2, where Y is the Young's modulus, r is the density, and l is the interatomic spacing. The table in HL p. 80 gives typical values of Y and r for solids, and you should know typical values for interatomic spacings from modern physics. Using the formula from class, calculate approximate values of the highest and lowest normal mode frequencies for a salt crystal 1 cm on a side. NaCl shows a strong absorption at around 60 mm wavelength; is this consistent with your approximate calculation?
Reading: For Friday, please read HL Ch 4 and Griffiths section 9.1.1.
1. HL Chapter 4, Problem 2.
2. HL Chapter 4, Problem 8.
3. HL Chapter 4, Problem 9.
Reading: For Monday, please read HL Ch 4.4-4.6.
1. HL Chapter 4, Problem 4.
Hint: The integral from -infinity to +infinity of the Gaussian
exp [-ax^2] is sqrt(pi/a). To get the integral of even powers
of x times this Gaussian, differentiate under the integral sign.
2. HL Chapter 4, Problem 11.
Reading: For Wednesday, please read HL Ch 2.5-2.7.
1. HL Chapter 2, Problem 11.
2. HL Chapter 2, Problem 12.
3. HL Chapter 3, Problem 4. Note also that the phase velocity
of these electromagnetic waves is greater than c. Does this constitute
a problem with relativity? Explain.
Reading: For Friday, please read HL Ch 13.
1. Find the Fourier coefficients for a sawtooth wave of unit
amplitude and period T, f(t) = (2/T) t, from -T/2 to T/2.
2. HL Chapter 13, Problem 3.
Reading: For Monday, please read HL Ch 13.4.
1. HL Chapter 13, Problem 1
2. HL Chapter 13, Problem 2.
3. HL Chapter 13, Problem 4.
Reading: For Wednesday, please read HL Ch 5.1-4.
1. HL Chapter 5, Problem 2.
2. HL Chapter 5, Problem 3.
3. HL Chapter 5, Problem 4.
Reading: For Friday, please read HL Ch 5.5-6.
1. HL Chapter 5, Problem 6.
2. HL Chapter 5, Problem 9.
3. HL Chapter 5, Problem 11.
Reading: For Monday, please read HL Ch 6.
1. HL Chapter 6, Problem 5.
2. HL Chapter 6, Problem 6.
3. HL Chapter 6, Problem 10.
Reading: For Wednesday, please read HL Ch 7.
1. HL Chapter 6, Problem 7.
2. HL Chapter 7, Problem 4.
3. HL Chapter 7, Problem 8.
Reading: For Monday, please read HL Ch 8.
1. HL Chapter 8, Problem 1.
2. HL Chapter 8, Problem 4.
3. HL Chapter 8, Problem 5. (Also try the experiment in problem
6 in your sink.)
Reading: For Wednesday, please read HL Ch 9.1-9.3.
1. HL Chapter 9, Problem 3. Note you have a second loop, which
means you must apply Kirchoff's current and voltage laws twice
each, rather than once. Remember what you are ultimately interested
in is relationships between I(x), I(x+Dx),
V(x), and V(x+Dx), so don't get sidetracked.
(Challenging)
2. HL Chapter 9, Problem 4.
Reading: For Friday, please read HL Ch 9.4-9.6.
1. HL Chapter 9, Problem 7.
2. HL Chapter 9, Problem 8.
3. You want to use a solar sail to travel from Earth to Mars "on
the cheap." Your spaceship has a mass of 1000 kg, including
your 1.0 km square aluminized Mylar solar sail. You orient your
solar sail so its surface normal points at the sun.
(a) What acceleration does your ship experience due to the light
pressure.
(b) Assuming that acceleration is constant, how long would it
take you to get to the orbit of Mars?
N.B. The intensity of solar radiation at the earth is 1300 W/m2, and the solar sail reflects all the
incident sunlight back toward the sun. For planetary data, check
HRW.
Reading: For Monday, please read Griffiths Ch 9.2.
1. *HL* Chapter 9, Problem 13.
2. Griffiths Chapter 9, Problem 9. (In Griffiths, the polarization
is the direction of the electric field vector. To sketch the waves,
just draw a few planar wavefronts on an x-y-z axis system.)
3. Griffiths Chapter 9, Problem 11. (Note the angle brackets <fg>
means the product is time-averaged. Remember f* is just the same
as f with all the i's changed to -i's.)
Reading: For Wednesday, please read Griffiths Ch 9.3.1-2.
1. Griffiths Chapter 9, Problem 13.
2. Griffiths Chapter 9, Problem 14.
Reading: For Friday, please read Griffiths Ch 9.3.3.
1. Griffiths Chapter 9, Problem 15.
2. Griffiths Chapter 9, Problem 16.
Reading: For Monday, please read Griffiths Ch 9.4.1-2.
1. Griffiths Chapter 9, Problem 18. (Hint: For (c), lambda
and v are MUCH different from their values in vacuum.)
2. Griffiths Chapter 9, Problem 19. (Hint: See p. 286 for conductivity
data. For (c), think about how a phase shift would show up in
a complex exponential.)
3. Griffiths Chapter 9, Problem 20. (Hint: For (b), remember the
quick and dirty way to get power from energy density for mechanical
waves on a string.)
Reading: For Wednesday, please read Griffiths Ch 9.4.3.
1. Griffiths Chapter 9, Problem 21. (Hint: Note beta >>
1, so use binomial approximations to get R. Don't just use the
expression from class--derive it!)
2. Griffiths Chapter 9, Problem 23. (Hint: Your values for A and
B should be within a factor of 10 of the literature values.)
3. Griffiths Chapter 9, Problem 24.
Reading: For Friday, please read Griffiths Ch 9.5.
1. Griffiths Chapter 9, Problem 27.
2. Griffiths Chapter 9, Problem 28. (Hint: The driving frequency
is given in Hz, so it's f, not w. You
need to convert your mode cutoff frequencies from w
to f to compare.)
3. Griffiths Chapter 9, Problem 30.
Reading: For Monday, please read HL Ch 9.7.
1. HL Chapter 11, Problem 1.
2. HL Chapter 11, Problem 2.
2. HL Chapter 11, Problem 3.
3. HL Chapter 11, Problem 4. Hint: I'd use the complex exponential
approach for this problem, rather than phasors.
Reading: For Wednesday, please read HL Ch 11.1-11.4.
1. Four narrow slits are cut in an aperture. As measured from the bottom slit, the positions of the other three slits are 10, 30, and 60 mm away, respectively. The slits are illuminated with a plane wave of coherent blue-green light of wavelength 500 nm. The resulting interference pattern is observed on a screen 1.0 m away. Use (a) complex exponentials and (b) phasors to derive equivalent expressions for the intensity of light as a function of position x on the screen (defining x=0 as the point where a normal to the aperture plane at the bottom slit intersects the observation screen). (c) Using Microsoft Excel, make a plot of the intensity as a function of x. Choose an appropriate separation between points where you evaluate intensity and an appropriate number of those points so that the plot clearly shows several interference fringes.
2. HL Chapter 11, Problem 5.
3. HL Chapter 11, Problem 13.
Reading: Reading relevant to these problems may be found in HL Ch 11.5-6. For Wednesday, please read the xeroxed handout (I'll put it on the bench outside my office Monday afternoon).
1. HL Chapter 11, Problem 7.
2. HL Chapter 11, Problem 9.
3. A transparent film has a thickness of 32.50 um and a refractive
index of 1.4000. Find (a) the number of half wavelengths m at
q=0 and (b) the first four angles at
which red light of wavelength 650 nm will form bright light fringes.
4. A thin film has a thickness of 465.0 um and a refractive index
of 1.5230. Find the angle f at which
dark fringe 122.5 will be observed if monochromatic light 656.0
nm is used as an extended source.
Reading: For Friday, please read HL Ch 11.6; EH Ch 10.1-10.4, 10.6.
1. Show that the secondary maxima of the Fraunhofer single slit intensity pattern are given by the roots of tan g = g, where g = ka/2 sin q. Show that these occur at approximately half integeral multiples of p, excluding +- 1/2 (hint: a graph might help).
2. In a single slit diffraction pattern, the intensity of the
successive bright fringes falls off as we go out from the central
maximum. Approximately which fringe number has a peak intensity
that is 0.5% of the central fringe intensity?
3. In the Fraunhofer diffraction pattern of a double slit, it
is found that the fourth secondary maximum is missing. What is
the ratio of slit width a to slit separation d? (This is essentially
a Physics 132 problem.)
Reading: For Monday, please read HL Ch 11.7, EH Ch 10.5, 10.7-10.10
1. We saw in class that the central intensity of the Fraunhofer diffraction pattern for a rectangular aperture was given by I = C2 a2 b2 (sin gx/gx)2 (sin gy/gy)2, where C2 = y2/l2R2. This expression for I is proportional to the square of the aperture area. Account physically for the fact, given that the power in the wave incident on the aperture is, of course, proportional to the first power of the area. Assuming the incident wave is a plane wave and the observation point is very distant, integrate the total power in the diffraction pattern to show that it matches the incident power. The integral from -infinity to +infinity of (sin g/g)2 is p.
2. Show that there are 1+2d/a maxima under the central diffraction envelope of a double slit pattern, where d is the slit separation and a is the slit width.
Reading: For Monday, please read EH Ch 10.5, 10.7-10.10
1. A diffracting aperture is in the form of an annulus with inner radius b and outer radius a. Obtain the expression for the intensity in the Fraunhofer diffraction pattern. Explain why the resolution is increased by adding the central disk. (Note, though, that the contrast is reduced because the intensity of the secondary maximum is increased.)
2. Compare the angular resolution of the human eye (dilated in
dim light) with that of an astronomical telescope of diameter
60 cm, for light in the middle of the visible spectrum. If a double
star system consists of two stars similar to the Sun with a separation
equal to the Earth-Sun separation, at what distance could they
be resolved in each case? (In general, larger Earth-bound telescopes
do not achieve higher resolution because of atmospheric turbulence.
However greater size is important for gathering light from very
faint stars. And adaptive optics with laser guide stars and deformable
mirrors, make it possible to avoid most of the effects of atmospheric
turbulence.)
3. Printers create letters as an array of dots on the printed page. Calculate the maximum dot to dot spacing needed so that a typical reader would perceive the letters as smooth and continuous. A typical laser printer does 300 dots per inch. How close would you have to be to the paper to discern the individual dots? (Note that in this analysis, you must neglect the flow of ink away from the point where it is applied; that flow smears the dots.)
Reading: For Wednesday, please read EH Ch 9.3-9.4, 10.1-10.2.
1. For plane waves normally incident on an aperture, the Fresnel-Kirchhoff diffraction integral is
y(P) = -i y0 / (2l) times the integral over the aperture of da (1 + cos q) exp [ikr] / r,
where r is the distance from the Huygens secondary wave source to the observation point P, and q is the angle between the normal to the aperture and a line from the center of the aperture to P. Show that in the Fraunhofer limit this integral reduces to the expression I asserted in class:
y(P) = -i y0 exp [ikR] / (lR) times the integral over the aperture of dx dh exp [-ik(ax + bh)],
where R is the distance from the center of the aperture to P, a = x/R, and b = y/R. Under what circumstances is this expression valid?
2. Imagine that you have set up the spatial filtering lab and that you want to create a uniformly illuminated circular diffraction pattern in the focal plane. What sort of aperture would you need to use, and how would its properties vary with radius?
3. The Grinnell orchestra is tuning in a concert hall, playing
a 450 Hz note. A long distance from the stage, there are two doors
leading out into a courtyard. Each door is 1.0 m wide, and they
are separated by 3.0 m center-to-center. Calculate and plot the
relative intensity of sound as a function of position at the opposite
end of the courtyard, 30 meters away from the doors. Take the
speed of sound in air as 300 m/s
Reading: For Monday, please read EH Ch 11.2-11.3, HL 11.8.
1. The on-axis diffraction by a circular aperture can be treated neatly with a construction using Fresnel zones. Divide the aperture into a set of concentric annular areas, choosing the outer radius rn of the nth zone such that the sum of the source and observation distances ro+rs is n half-wavelengths greater than the distance Ro+Rs through the center. The Fresnel-Kirchoff diffraction integral can now be expressed as a discrete sum over the contributions from exposed zones. Moreover, the contribution from adjacent zones are out of phase by p and interfere destructively with one another.
(a) Show that the outer radius of the nth zone is rn = sqrt [nlRoRs/(Ro+Rs)] for r2 << Rs2, Ro2.
(b) Show that the area of the zones is constant.
(c) Assuming that the angles qo and qs are small, show that the intensity is a constant maximum for an aperture exposing an odd number of zones, and it goes to zero for an even number. (Hint: use a phasor representation.)
(d) What is the intensity on the axis of a zone plate in which alternate zones are transparent and opaque? Assume there are N transparent zones. (Think phasors again.)
(e) Show that a zone plate focuses a point source at a distance o onto a point image at a distance i, where o and i are related by the thin lens equation 1/o + 1/i = 1/f = l/r12, where r1 is the outer radius of the first Fresnel zone. (You need to use your result from part (a)).
Reading: For Monday, please read EH Ch 11, HL Ch 11.8.
1. In a diffraction experiment, a point (pinhole) source of wavelength 600 nm is to be used. The distance from the source to the diffracting aperture is 10 m, and the aperture is a hole of 1 mm diameter. Determine whether Fresnel or Fraunhofer diffraction applies when the screen-to-aperture distance is (a) 1 cm, (b) 2 m, (c) 10 m. Hint: a criterion to use here is whether the maximum Fresnel phase difference across the aperture d = p/2 * [2/l * (1/Ro + 1/Rs)] * a2 is small compared with p.
2. Light from a HeNe laser (l = 632.8
nm) is diverged by a microscope objective, passes through a pinhole
spatial filter, then travels 30 cm to a razor blade. A screen
is placed 160 cm behind the razor blade. Using the Cornu spiral
(handed out in class or available for pickup outside my office),
generate a graph of intensity as a function of position on the
screen. The graph can be either hand-drawn or plotted using a
computer. Note: since one edge of the aperture is at -infinity,
you may take S(-infinity) = C(-infinity) = -1/2 as the tail of
the phasor in all cases. The key here is to convert the position
of the observation point x to a dimensionless u coordinate along
the Cornu spiral, which will be the head of the phasor for that
position.
1. Determine approximately the ratio of the probability of spontaneous emission to the probability of stimulated emission at room temperature in (a) the x-ray region of the electromagnetic spectrum, (b) the visible region, and (c) the microwave region.
2. At the heart of a HeNe laser is an electrical discharge tube
known as a plasma tube. When a high voltage is placed across the
plasma tube, many atoms are excited, and those excited atoms emit
electromagnetic radiation at characteristic frequencies, including
the lasing frequency f0. But
the atoms are all moving around and have a velocity distribution
in the x-direction (the axis of the plasma tube) P(vx)
= sqrt [m/2pkT] exp [-mvx2/2kT],
where m is the mass of an atom, k is Boltzmann's constant, and
T is the absolute temperature. Due to this motion, there is a
nonrelativistic Doppler shift in the frequencies observed in the
rest frame of the plasma tube. The nonrelativistic Doppler shift
of radiation emitted in the x-direction is f = f0(1
+ vx/c). The resulting wavelengths
observed in the laser are spread to higher and lower values due
to the (respectively) lower and higher frequencies corresponding
to negative and positive values of vx.
We say that the spectral line has been "doppler broadened"
(this is what allows us to see the laser lines easily in a spectrometer,
since the Heisenberg uncertainty principle does not cause significant
line broadening in atomic transitions).
(a) What is the mean frequency <f> of the radiation observed in the plasma tube?
(b) To get an idea of how much a spectral line is broadened at particular temperatures, derive an expression for the standard deviation of frequencies, defined to be standard deviation = sqrt [(f - <f>)2]. Your result should be a function of f0, T, and constants.
(c) Use your result from (b) to estimate the fractional line width, defined by the ratio of the standard deviation to f0, of neon gas at T = 350 K.
1. Laser light (l = 500 nm) is diverged by a microscope objective, passes through a pinhole spatial filter, then travels 2.00 m to a slit 2.0 mm wide. A screen is placed 2.00 m behind the slit. Using the Cornu spiral (handed out in class), generate a graph of intensity as a function of position on the screen. The graph can be either hand-drawn or plotted using a computer.
2. Show that the matrix equation given in class:
|
= |
|
|
|
|
|
|
can be simplified to the expression
|
= |
|
|
where g1 = 1 - L/R1 and g2 = 1 - L/R2.
3. A confocal resonator is a pair of spherical mirrors arranged along a common optical axis in such a way that their focal points coincide. Evaluate the stability of a symmetric confocal resonator (where the two mirrors have the same radius of curvature) using the matrix stability criteria discussed in class. What are the eigenvalues and eigenvectors.
Halliday, Resnick, and Walker, 5th Edition, Chapter 35, problems 17, 18, and 54.
Halliday, Resnick, and Walker, 5th Edition, Chapter 35, problems 13, 19, and 37.
1. The Apollo 11 astronauts left a corner cube reflector on the surface of the moon. This reflector has been used to measure precisely the earth-moon separation by a time-of-flight technique, similar to the technique you used in modern physics lab to measure the speed of light. Suppose that you send a powerful laser pulse toward the moon. If the beam has a TEM00 Gaussian cross-section and a 1 cm waist at the laser, how big is the laser beam when it reaches the moon?
2. A HeNe laser exactly 25.0 cm long is lasing at a wavelength
l = 632.80 nm in the TEM00
mode. What is (a) the number of antinodes in the standing wave
pattern in the cavity and (b) the frequency separation between
modes?
3. The beam emerging from a diode laser is astigmatic: it diverges at different angles in the horizontal and vertical directions. For typical laser with wavelength 850 nm and active area dimensions 0.2 µm horizontally x 500 µm vertically, calculate the asymptotic horizontal and vertical angles of divergence. At what distance from the laser is the beam circular?
Reading: For Monday, please read HL Chapter 10, which lays out a nonrigorous approach to radiation. You may also wish to scan Griffiths Chapter 11 so you see what is involved in calculating radiation potentials rigorously.
1. HL Chapter 10, Problem 1.
2. HL Chapter 10, Problem 4.
3. HL Chapter 10, Problem 5.
Reading: For Wednesday, please review HL Chapter 11.4, and read Griffiths pp. 443-465.
1. HL Chapter 10, Problem 3.
2. HL Chapter 11, Problem 18.
3. HL Chapter 11, Problem 19. Hint: expand the exponential phase factor before integrating