Single and Double slits -- the difference for first-year physics students

from wikipedia's diffraction article

The double slit is covered in detail on the AP Physics B exam.  Students should be able to explain why an interference pattern is formed, cite an equation for the positions of the maxima, tell a bit about where that equation comes from, and describe qualitatively how the brightness of the pattern varies along a screen.

The single slit is not covered in quite so much detail.  Good thing, too.  I spent some time last week learning the details about single slit diffraction.  You see, one of the problems for the USIYPT next year is about a pinhole camera.  To understand the image formed by a circular slit, one first has to understand how a one-dimensional single slit works.

At the Physics B level, we have to know that the positions of single slit minima are identical to the positions of double-slit maxima. That is, the relevant equation x=mλL/d for double slits gives bright-spot positions for whole number m's; for single slits, the same equation gives dark-spot positions for whole number m's.  The central maximum for single slit diffraction is much wider than for double slits.

Why?  I don't go into that in AP Physics B.  I tell the convenient mistruth that diffraction occurs simply because light diffracts around the edges of the slit, and interferes similarly to the double slit situation. The true reasoning is very, very complicated. 

At a deeper-than-introductory level, the single slit can be considered as an infinite set of point sources.  Constructive interference occurs at the central maximum.  Diffraction minima can be found by trigonometric analysis of when each of the infinite point sources can be paired with another whose path difference to the location on the screen is λ/2.  The diffraction maxima are not halfway between the minima; rather, their positions are determined by the solution to a complicated differential equation whose solutions are Bessel functions. 

(Confused?  Me too.  I spent a couple of class periods and finally understood the derivation of the minima, but the maxima derivation is still flummoxing.)

This year's operational (i.e. regular form, not "form B") AP Physics B exam asked about a single slit.  Students were to design an experiment to measure the width of the slit.  The fact that it was a single, not double, slit was nearly irrelevant in this case.  Using x=mλL/d still works, as long as x and m are described appropriately.  Measuring the distance between dark spots with m = 1 works just fine whether this is a single or double slit.

The Form B exam, though, really hit at the qualitative difference between the single and double slit.  It asked to graph the intensity as a function of location on a screen for a double slit; then for a single slit under the same conditions.  Answer:  the central maximum remains, double-slit-maxima become single-slit-minima. 

And  this is as deep as a Physics B question will get about single slits.   

GCJ

Thickness of Gray vs. Brown Hair

In my AP physics institutes, I show the classic demonstration in which we measure the thickness of a hair via diffraction.  I hold the hair in front of a green laser, and project the resulting diffration pattern on the white board.  We measure the distance x between dark fringes. 

The thickness of the hair d is given by the equation x = mλL/d, where m = 1 (because we measured between two fringes, not across several of them), λ is the wavelength of the laser, and L is the distance to the screen.

The thickness of every hair I've ever measured has been between 10 μm and 100 μm.  Though I've had students investigate, we've never found any significant and consistent difference in thickness between different colored hair, between boys' and girls' hair, or between "thick" and "thin" hair.  Perceived "thick" hair has the same diameter as anyone else's hair, it must just lie together differently so that the distance between hairs is larger.

At my Manhattan College institute, I asked for a volunteer hair for this demonstration.  I looked around the room and saw virtually everyone with either bald heads or extra-short, nearly shaved, haircuts.  Fortunately, Manhattan College alumna Regina Verdeschi shook out her hair clip to reveal a deep brown mane that hung halfway down her back. 

Regina asked, "Do you want a brown hair, or a gray hair?"  I froze for a moment.  For one thing, no student of mine has EVER asked that question.  But more to the point, with my wife Burrito Girl that's a trick question to which I know the answer:  "Brown, of course, because you don't have any gray hair."  (This will be the correct answer even in 20 years when her hair is deepest silver.)

Even though Regina is emphatically NOT Burrito Girl, I still gave the appropriate answer: "Brown, please.  I didn't notice any gray hair."  I've been trained well, I guess.  We measured the thickness of Regina's brown hair to be about 50 μm.

When I described my history of 10-100 μm hair thicknesses in every color and gender, Regina spoke up again.  "What about gray hair?  It's noticably thicker.  That's why I asked."  She proceeded to pull one of her few gray hairs and hand it to me.  She was right that the hair felt noticeably thicker. 

At this point, the entire class was clamoring for a second measurement.  We projected the diffraction pattern and saw immediately that the fringes were considerably closer together.  The calculation suggested that Regina's gray hair was about 75 μm thick, or 30% thicker than her brown hair.

While I will not be able to repeat this experiment in my class of 16-18 year old boys, some of whom are balding but none of whom is going gray, I would be interested in feedback from the audience.  Measure one of your colleagues' gray hairs.  Tell me your results.  *Is* gray hair generally thicker than not-gray hair?  I'd do the experiment myself, but neither I nor Burrito Girl has access to the necessary materials.