Mathematics-Physics High School Meeting
14 March 2000
Notes Taken by Porter Johnson

John Bozovsky (Bowen HS)
treated us to A Physics Mystery!
Using a tape player (which had crisp,loud sound), he played a cassette of an Ellery Queen mystery. John enumerated the cast of characters on the board for us, and also critical questions for us to think about:

When the tape was done (we tend to forget how interesting the old radio stuff could be!), John pointed to each name and asked us who thought that person was guilty; then he recorded our vote. It turned out that Cornelius van Clique (from Holland) received the most votes - and it turned out that was correct! As John explained, the guilty person had dug in the wrong place because he assumed the map referred to distances in meters, and it was actually intended in yards. The only person from the continent, and who would normally assume meters, was the man from Holland. The measurements that had been paced off were wrong by about 1 part in 10, just about the difference in length between meter and yard. Neat!

Comment by PJ: The British fought 5 "marine trade wars" with the Dutch in the 18th century, and lost them all, because the Dutch had better and faster ships. As a result, anti-Dutch expressions from British English are fairly common: i.e., "Dutch courage", "Dutch uncle", and "Double Dutch".

Bill Blunk (Joliet Central HS)
set up the Millikan Oil Drop Experiment on the table. It is a dandy piece of equipment sold by Sargent Welch, and expensive, so his school could afford only one, Bill explained. So when he sets it up for his students, only one at a time can look through the telescope to see the oil drop(s).

He then showed us a new addition to his technology - a small video camera that he had bought for $90 at the ISPP meeting at New Trier HS. (It's the sort of thing being used on computers these days when people are "talking" to each other.) It was now connected to a large TV set in front of us, and when Bill aimed the camera at us, we could see ourselves on the TV.

He reviewed for us how the Millikan Oil Drop Experiment works; a pair of horizontal, parallel conducting plates are placed about 1 cm apart, and an electric harge is placed on them. Then some "oil drops" are squirted into the space between them (using an atomizer with a hollow needle such as for inflating a basketball).

Some of the drops become charged and may have 1, 2, 3, etc electrons on them. (Millikan used oil drops because he found small water drops evaporate rapidly, oil drops don't.) With the aid of a dandy diagram on the board which showed a charged sphere and a rod nearby, Bill showed us how opposite charges attract and repel. He used colorful magnets that had the + and - charge signs on them. They stuck to the board on the diagram and Bill could move them around to show how charges respond to each other -- a la Bill Shanks.

Bill Blunk also explained that nowadays fairly uniform latex spheres averaging 913 nm in diameter and carried by water drops from the atomizer are what he squirts into the space between the plates. A sphere (drop) with one electron negative charge would be attracted toward the upper positively charged plate. If a drop had 2 electrons and twice the negative charge (assuming they are all alike), then it would move twice as fast. By observing the motion of the drops through the telescope against a reticule (grid), one could calculate their speeds.

At this point, Bill placed the video camera to "look" right into the telescope, and we could then see the drops on the TV screen! With the voltage off (no charge) the drops would gradually move upward (which was really down, since the telescope inverts the image) under gravity. But with the voltage on, some would move down (actually, up, as seen on the TV!). But they moved with different speeds, and the differences between their speeds was always the same amount, which means that the electron charges on the drops always differed by the same amount. Bill could now show this to the entire class at once with the aid of his new video camera. Great! And it is affordable!

Roy Coleman (Morgan Park HS)
told us that today was p Day! (Today's date: 3.14) And Albert Einstein's birthday! And then he shared some information about the AP Physics Conference at Triton College.

Porter Johnson (IIT Physics)
explained Coffee Cup Caustics to us with some mathematical detail, but nicely presented.

The Coffee Cup Caustic is shown on the website, on which the reflected image from a real coffee cup is shown, as well as a "Monte Carlo" simulation of the event.

Let us suppose that rays parallel to the y-axis strike an upper semi-circle of radius R from the inside, and are reflected. If we let the angle between the reflection point be q, the coordinates of the point of reflection are x = R cosq and y = R sin q. The angle of incidence of the ray, relative to the normal, is p/2 - q, and the angle of reflection has that same value, as well.

The reflected ray travels at an angle p/2+2 q relative to the positive x-axis, and strikes the circle again at a point an angle 3 q from the horizontal, at the point (x = R cos3 q, y = R sin 3 q). The equation for the reflected ray is

y - R sin q = (x - R cos q) tan (p/2 + 2q).
y = R sin q - (x - R cos q) / tan 2q.

As q varies, we generate a series of straight lines. To find the envelope of those straight lines, we must determine the maximum value that y can have for a given value of x, and the appropriate choice of q, by setting the derivative dy/dq equal to zero. Thus we obtain

dy/dq = R cos q - R sin q/ tan 2q + 2 (x - R cos q)/ sin2 2q = 0.

Thus, we obtain the relation
x = R (3 cos q + cos 3 q ) / 4 = R cos3 q.
y = R sin q - (x - R cos q)/ tan 2q = R sin q (1/2 + cos2 q ) ,
y(x) = R ( 1/2 + (x/R)2/3 ) Ö (1 - (x/R)2/3 ).

The reflected straight path runs between two points on the circular rim, corresponding to angles q and 3 q with respect to the x-axis. These two points on the boundary have coordinates (x, y) = (R cos q, R sin q) and (R cos 3 q, R sin 3 q), respectively. Each such straight line is tangent to the "caustic curve" at one point, which lies precisely one fourth of the way along the path, with coordinates

x = R (3 cos q + cos 3 q )/4
y = R (3 sin q + sin 3 q )/4

We cycle through the various striking points by letting q vary between 0o and 180o, or p radians. For each value of q, we obtain a point on the envelope, or caustic curve.

A template of a 360o Protractor was handed out, and participants made their own caustic by drawing lines from the positive x-axis [right on the middle] from angles q to 3 q. Here is a set of angles relative to the horizontal to use:

10o --- > 30o
20o --- > 60o
30o --- > 90o
40o --- > 120o
50o --- > 150o
60o --- > 180o
70o --- > 210o
80o --- > 240o
90o --- > 270o

The right side of the caustic will arise, as if per magic!

The full curve, which was produced using the software package EXCEL, is given below. An excellent reference on using EXCEL in graphics is the book EXCEL for Scientists and Engineers by William J Orvis, Second Edition [ISBN 0-7821-1761-9].

Maybe someone can show us a real, live caustic at our next meeting?!
Some really good math and Physics Phun!!