High School Mathematics-Physics SMILE Meeting
27 March 2001
Notes Prepared by Porter Johnson

Don Kanner (Lane Tech HS, Physics)
drew the following temperature versus time graph for a fixed volume of water, held at atmospheric pressure, with a constant power into a system:

He remarked that his students were able to compute the relative times for the stages A, B, C, D, and E, but that they were totally unable to design an experiment to gather data.  When asked to sketch an appropriate system of  apparatus, students would typically show a block of ice with an energy source acting upon it, but not one student included a thermometer, or showed a vessel for containing water.  During stages C, D and E, when water vapor is produced, one would need something like a "weighted piston" to maintain constant pressure, while permitting the volume to expand significantly---perhaps by a factor of 1000.  His experiences illustrate the need to include experiments on thermal physics in the standard high school physics curriculum.

Don also mentioned a self-checking graph, associated with the Toilet Flushing Experiment designed circa 20 years ago by Roy Coleman.  Working in pairs, students were asked to flush a toilet, and to record the depth of water in the reservoir behind the toilet seat, as a function of time, in intervals of roughly two seconds. Most students got a graph like that appearing on the left below, which resembles a check mark.  Upon occasion, student teams would obtain a graph like the one on the right below.  Those students, who had not followed instructions properly, were measuring water depth in the wrong chamber!

Fred Schaal (Lane Tech Park HS, Math)
raised the following questions:

Larry Alofs (Kenwood HS, Physics)
obtained two sheets of polarizing material, along with a quarter-wave plate and a half-wave plate as a give-away at the 13 March 2001 ISPP meeting at the University of Chicago.   However, he wasn't sure how the wave plates should be used, and he brought them to the meeting for experimentation.  We learned that when the two polarizing sheets were placed at 90o to one another, no light could pass through them.  However, when one of the wave plates was placed between the polarizing sheets in that crossed position, light did pass through.  How come?  What is a wave plate? 

Comments by Porter Johnson:  Most crystals are optically anisotropic, in that the the velocity of light passing through the crystal depends upon the direction of polarization of the electric field vector E.  Let us consider normal incidence of light [z-direction] upon a thin crystalline material such as mica or quartz, with different speeds for electric field in the x and y crystal axis directions, respectively.  In a quarter wave plate the phase difference for light polarized in the x- and y-directions is 90o, or a quarter wavelength.  Suppose that the light passes first through a polarizer, becoming linearly polarized in the process. Then let the light pass through a quarter wave plate that is tilted at 45o to the direction of the polarized light.  Relative to coordinates in the quarter wave plate, the light is a superposition  of light with electric fields in phase [for each photon, at least] with equal components in the x and y directions.  The quarter wave plate produces a phase difference of 90o for the y-polarized component relative to the x-polarized component, resulting in circularly polarized light.  A schematic diagram, taken from the website http://www.meadowlark.com/AppNotes/appnote1.htm, is given below:


Circularly polarized light is a coherent superposition of the states of linear polarization, with equal weights. Therefore, when we put a second polarizer behind the quarter wave plate, we observe 50% transmission if the second polarizer is parallel to the first, and 50 % transmission if the second is perpendicular to the first.  We should thus observe half intensity for both cases.

In a half-wave polarizer, the relative phase in the crystal axis directions is shifted by 180, so that, in effect, Ex ® Ex and Ey ® - Ey. Thus, if the crystal axis is placed at 45o to the initial direction of polarization, the beam will be polarized at 90o to its initial direction after leaving the crystal.

Here is a description, from that same source, of the effect of a half wave polarizer, in which the phase difference between x-and y-polarized waves is 180o:

"A simple polarization rotator consists of a half wave plate in linear polarized light. Rotating the half wave plate causes the polarization to rotate to twice the angle of the half wave plate's fast axis with the polarization plane, as shown in Figure 5A. We achieve variable polarization rotation by aligning the fast axis of a variable retarder at 45° to the incoming polarization and following this component with a quarter wave retarder with its slow axis aligned with the incoming polarization as seen in Figure 5B. The amount of rotation achieved depends on the amount of retardance exhibited by the first retarder. The polarization axis is rotated to an angle that is one-half the phase shift provided by the variable retarder."

The discussion then turned to the problem of production of electricity.

Energy Production in the United States, by Source Type can be found at: http://www.eia.doe.gov/emeu/iea/table29.html

The shortage of electricity is especially acute during periods of peak demand [such as hot summer days with high air conditioning load].  In particular, the problems of transmission of electricity over large distances is made more difficult by mounting resistive losses, as well as the fact that 60 Hz corresponds to a wavelength of light of about 5000 kilometers, so that any transcontinental transmission line would radiate significant amounts of electromagnetic radiation, since it would functon as an antenna.  Here is a discussion of various types of energy storage systems that might be used to meet the peak electrical demand:  http://www.eere.energy.gov/.

One type of system, called Pumped Storage Facilities, involve pumping water into a high reservoir at periods of excess electrical capacity [such as at night], and then making electricity through a hydro-electric system during periods of high demand.  Here is a photograph of the facility at Ludington Michigan

Source: http://www.aucmi.org/Resources/ludington.html

"The Ludington Pumped Storage Plant consists of an energy production facility and a 110-feet deep reservoir on Lake Michigan that holds 27 billion gallons of water at a height of 363 feet above lake level. The reservoir is lined with a innovative asphalt barrier to limit seepage. When the demand for electricity is low, water is pumped into the reservoir from the lake. Then when the demand for electricity is high, water is released from the reservoir, flows through penstocks and turns 6 reversible, 433,000 horsepower turbines to make electricity. The total output of 1,872 megawatts is enough to power a city of 1,400,000 people. Customers throughout Michigan use the energy generated here. The relatively simple energy production technology is environmentally benign as no fuels are burned and no pollutants are emitted."

Here is a schematic of the TVA facility on the Tennessee River just south of Chattanooga, at Raccoon Mountain Tennesseehttp://www.tva.gov/sites/raccoonmt.htm

Pumped Storage Facility Diagram on site http://www.tva.gov/power/hydro.htm

Information on the Queenston Ontario Pumped Storage Facility [on the Niagara River just below Niagara Falls] can be obtained at the website:  http://www.ewh.ieee.org/reg/7/diglib/library/electricity/pdf/P_one_5.pdf

In the planning process for the Deep Tunnel Project in the Chicago area, there was serious discussion of setting up a Pumped Storage Facility between the Chicago River and the Deep Tunnel [over 50 meters of altitude difference--more than Niagara Falls!].  Such a facility would be in an ideal location for meeting peak demand for electricity in downtown Chicago, and would have the added benefit that aspiration / aeration of  the water would add oxygen to it, combating eutrofication [http://www.emecs.or.jp/99cd-rom/file/chap2/kankyo/eiyo-e.htm] and, in effect, purifying the river water.  The proposal was generally felt by technical experts to be an excellent idea, but it was never implemented.  It could still be built, and the reasons are more compelling than ever!  

Here is some information on the Chicago Water Reclamation District Tunnel and Reservoir Project [TARP]

It was suggested that geothermal and other renewable energy resources should be given more serious consideration as sources of energy.  In particular, tidal energy is a potential source.  The greatest tides on earth are found in the Bay of Fundy, which lies between the Canadian provinces of New Brunswick and Nova Scotia, just North and East of the State of Maine.  The typical tide height differential there is about 20 meters.  It produces the Reversing Falls / Les chutes reversibles on the ST John River, in ST John, New Brunswick.  See the officially bilingual website http://new-brunswick.net/Saint_John/reversingfalls/reversing.html.

See you next time!

Notes taken by Porter Johnson.