High School Math-Physics SMILE Meeting
01 May 2001
Notes Prepared by Porter Johnson

Lee Slick (Morgan Park HS) Handout:  English Units (only for non-metric die-hards!)
gave the following information about English units in excruciating detail:

1 league = 3 miles 1 nautical mile = 1.154 miles 1 Roman mile = 0.949 miles
1 mile = 8 furlongs 1 furlong = 10 chains 1 chain = 4 rods
1 mile = 5280 feet 1 foot = 12 inches 1 cubit = 18 inches

2 mouthfuls = 1 jigger 2 jills = 1 cup  2 quarts = 1 pottle 2 pails = 1 peck
2 jiggers = 1 jack 2 cups = 1 pint 2 pottles = 1 gallon 2 pecks = 1 bushel
2 jacks = 1 jill 2 pints = 1 quart 2 gallons = 1 pail  ... etc ...

Fred Farnell (Lane Tech HS, Physics)
reflected a laser beam off a mirror [with silvering on the "back surface" behind a glass layer] and we saw 4 reflected laser "spots" on a screen.  The spots were produced by reflection off the front and back surfaces of the glass layer, as well as multiple reflections, and it was suggested that there might be additional spot images with lower intensity.

Then he reflected the laser off a white plastic surface.  It was hard to see any image on the screen; the "spot" was very diffuse and spread out.

Finally he reflected the laser off the shiny side of a piece of aluminum foil, and the resulting smeared image resembled a pattern from a diffraction grating. In fact, one could see lines in the foil by close examination.  When the foil was turned by 90°, the smeared image was also rotated by that amount.  Upon repeating the experiment with the dull side of the foil, two-dimensional diffraction patterns were found.

Larry Alofs suggested that thin plastic layers are put on both sides of certain types of  aluminum foil, and it was speculated that this might be to reduce oxidation. For more information on Aluminum foil, see the website http://www.aluminum.org/Content/NavigationMenu/TheIndustry/Foil/Foil1.htm.

Fred Schaal asked how long it takes copper sheets to form the (green) oxidation layer.  According to Porter Johnson, the time scale is about 18 months, if the copper sheets are exposed to the elements. For details see http://www.paradigmshingles.com/weathering.html.

Arlyn Van Ek asked why an index of refraction occurs in a material, when light always travels between the atoms with velocity cPorter Johnson made the following comments.  A freely propagating plane wave of electromagnetic radiation can be written (leaving out its state of polarization) exp[ i k · r - w t] = exp [ i k r cos q - w t], where k is the wave vector [z-direction] and w is the angular frequency. In the presence of a spherical scatterer (a free atom) at the origin, there is also a scattered wave (spherical polar coordinates, with time factor suppressed):

exp[ i k z] ® exp[ i k z] + F(q) exp [i k r] / r

The function F(q) is called the scattering amplitude; in general it is complex and dependent on wave number k. The two terms are equally important for propagation of light in a medium, and we obtain the form

exp[ i n k z].

The index of refraction of the medium of (randomly located) atoms, n, is given in terms of the forward scattering amplitude F(0), and N, the number of atoms per unit volume, by the following formula:

n = 1 + 2p N F(0) / k2 ,

The imaginary part of index of refraction corresponds to attenuation of the beam.  The intensity transmitted to depth z is given by the relation

I(z) = exp[ - 2 k z Im n ] = exp[ - 4p N z Im F(0) / k] = exp[ - N z s] ;

where s = 4p Im F(0) / k is total scattering cross section. This latter relation is known as the optical theorem.

In summary, the index of refraction arises because of the scattered wave, which affects the amount of light passing through the material. For details see Scattering Theory of Waves and Particles by Roger Newton [McGraw-Hill 1966] pp 24-28.

Don Kanner (Lane Tech HS, Physics)
looked through a magnifying glass (lens) at a pencil that he held on the other side at a distance less than the focal length, f, of the lens. Don then told us how his students were mostly confused about where the image of the pencil would be, and how it would look. He told us that, according to optics, an erect, virtual and magnified image is formed on the same side of the lens as the pencil. How could we see it? By placing a screen at its location? But then he showed us the image by turning the lens-and-pencil so we could look through the lens and see the virtual image for ourselves. We knew it had to be a virtual image since it was larger than the actual pencil as seen without the lens. But some of his students were surprised that they could see only the virtual image through the lens, and not the pencil "itself" --- after all, the lens is transparent and they thought you should be able to see the "actual pencil"! Great ideas, Don

Marilynn Stone (Lane Tech HS, Physics) Polarization of Light
made a pinhole at the center of an opaque sheet of paper and placed it on the overhead projector. We could see a spot of light (pinhole image) on the screen. She then sandwiched the paper-with-pinhole between two sheets of transparent polarizing film. Rotating the top sheet caused the spot of light to alternately dim and brighten, which is what we physics teachers would expect. But then she removed the top sheet and placed a transparent crystal on top of the pinhole. Surprise! We now saw two spots of light on the screen, instead of one! How come? It turned out that the crystal was Calcite, which is birefringent, meaning it has two different indexes of refraction depending on the direction of polarization of the light passing through it. So the polarized light from the pinhole was refracted at two angles to form two distinct pinhole images! Then, as Marilynn rotated the crystal (about its vertical axis), the spots rotated about each other while one would brighten and the other would dim, alternately "winking" on and off. Beautiful!

Next, when she sandwiched a transparent plastic protractor between the two sheets of polarizing film, we saw the image of the protractor on the screen, but it had pretty "rainbows" of color throughout. This indicated internal mechanical stress frozen into the protractor, resulting in index-of-refraction variations to produce the colors. Ann Brandon said that such protractors had a maximum stress located near the 45° mark, due to the process of manufacturing, and tended to break there. Marilynn then put a transparent, U-shaped plastic piece between the films, and as she squeezed the ends of the U, she produced internal stresses which resulted in beautiful rainbow colors within the projected image of the U. Next, she produced a colorful display by placing a transparent sheet between the polarizing films. The sheet was covered with criss-crossed pieces of cellophane sticky tape. Can you explain this?

She put two quarter wave plates on top of one another, to make a half-wave plate, and them between the polarizers. When she rotated the plates, there were 4 cycles in a full rotation, but only 2 cycles when she rotated one of the polarizing sheets. The half wave plate at angle q to the beam polarization rotates that polarization by angle 2 q.

She passed around the reference book Polarization of Light - Basic Theory and Experiment by Hollis N. Todd,  [Bausch & Lomb Inc. 1970] pp 24-25, which described quarter wave, half wave, and full wave plates.

Bill Colson (Morgan Park HS, Math)
showed a number of different ways of writing expressions that are identities for the number 1:

-i2 = - ( -1 ) = e0 = sin2 q +cos2 q = 1

Porter Johnson mentioned the famous Euler relation

ei p + 1 = 0 ,

which involves the "five most important" numbers 1, 0, i, e, and p.

Bill Blunk (Joliet Central HS, Physics) (A Harald Jensen Presentation)
showed us a ceramic disk magnet (magnetic particles imbedded in ceramic material) that had a hole through its center, for which the front surface was the North Pole of the magnet.

Bill sketched the ceramic disk magnet on the board, with its north pole facing us, and sketched many current loops (representing the domains) and showing the current in each loop as counterclockwise, consistent with the face of the ceramic magnet being a north pole. With each loop drawn next to its neighbor, it was easy to see that their currents would would cancel out, except at the outside circumferential surface on the rim of the magnet, where the loops had no neighboring current loop for cancellation. Thus, there is a surface current I counter-clockwise as seen from the front, and a surface current -I (clockwise as seen from the front) would circulate around the inner hole.

Bill demonstrated the presence of the outer current loop by placing a small compass needle attached to a non-magnetic brass shaft hear the outer edge of the magnet, and showed that the field points outward.  Then, he placed the same needle near the inner edge, and showed that the field pointed in the opposite direction--inward.  Thus, the presence of the surface currents are detectable.  Very nice, Bill!

Of course, that is not the end of the questions one may ask. Why can we think of the domains as tiny current loops?  But we'll leave that for another day! (The answer lies with unpaired electron spins within the domains.)

You did Harald Jensen proud, Bill!

Richard Goberville  (Joliet Central HS, Physics) Resonance

Ann Brandon (Joliet West HS, Physics)

Arlyn Van Ek (Illiana Christian  HS, Physics)
brought in a magic liquid that appears to have the capability to restructure broken glass.  He took a Pyrex™ glass test tube, crushed it on the table, and  put the pieces into a beaker full of the magic liquid.  He then reached into the beaker and pulled out the unbroken test tube.  The magical properties of the liquid [Wesson Oil™ or Vegetable Oil, actually] are associated with the fact that its index of refraction is the same as glass, so that we could  not see the other (unbroken) test tube already inside the beaker, or the broken glass pieces remaining in the bottom of the beaker! Very sly, Arlyn!

Estellvenia Sanders (Chicago Vocational HS) Teeing for Angles
made a rectangle on the floor about 2 ft wide and 10 ft long using masking tape. She marked the tape at 1 ft intervals. She then gave each of three volunteers a toy plastic golf club and plastic ball. Each volunteer was asked to putt the ball to see the distance it would go before it either stopped or went out-of-bounds. A chart was drawn on the board, with each person's name displayed on the vertical-axis, and the distance on the horizontal-axis. Each distance was located as a dot on the chart. Straight lines were drawn to connect each pair of dots on the chart as data was obtained. The lines made various angles with each other, which the we were asked to identify as obtuse, acute, right angle, etc. A geometry vocabulary was thus motivated by this game: angle, point, plane, line, etc. Estellvenia uses hand signing to communicate with her deaf students, and this kind of activity proves quite helpful. Thanks, Estellvenia!

Roy Coleman [Morgan Park HS] Puzzle
handed out the following pattern on a sheet of paper:

What should go in the region in which ?? is located?  The answer is based upon the pattern for displaying numbers on hand-held calculators, which involves a display on which any of seven lines are present. The number 8 involves all seven lines whereas the number 4 has four lines and the number 7 three lines, as shown:

Roy's pattern shows a complementary image, in which only the "unlit" lines are shown. Let us put in the lighted lines by using RED lines:

Therefore the missing pattern is given by the GREEN lines in the last box.


Notes taken by Porter Johnson