What it shows:
When unpolarized light is reflected from a non-metallic surface, the reflected ray is plane polarized parallel to the reflecting surface if
θi + θr = 90°
tanθi = n
where θi = incident ray (Brewster's angle), θr = refracted ray, n = refractive index
How it works:
We use a black vinyl sheet 1m×4m as the reflecting surface, which has a Brewster angle of 57°. A theatrical spot lamp 1 is used to give a 50cm circle on the plastic and a good meter diameter spot on the wall of the lecture hall. The lamp is fitted with a large protractor and plumb line to show the angle of incidence to the audience. Illuminate the sheet at angles other than Brewster's, and hold the Polaroid between the lamp and the vinyl. Rotate to show that the spot on the wall dims, but is never extinguished. Adjust to 57° and again rotate the Polaroid to see the spot vanish when the Polaroid's axis is vertical.
Setting it up:
Place the vinyl on a cart with its long axis across the hall and the spot lamp at one end, looking down from about a meter above. You'll want the setup to give a spot on the far wall that's nice and clear at 57° (obviously it's going to move up and down the wall as the incident angle is changed). Use a large Polaroid sheet; (0.5m × 0.5m is good) with the axis of polarization clearly marked.
You need all the hall lights down to see this effect clearly. Choose comparison angles far from the Brewster angle (30° and 75° say) so the spot definitely doesn't vanish.
1 Keg-Lite, by Bardwell & McAllister, Hollywood, CA