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28 July 2012

Ray Diagrams - templates for geometric optics

In an Honors or AP course, lenses and mirrors are covered both quantitatively and qualitatively.  This means that students must be able to draw ray diagrams to show the location and size of an image; AND they must be able to calculate image position and size using equations.  

The trick to teaching geometric optics at this level is to ignore the calculational piece.  Let the students figure out on their own how to use the calculator to solve the thin lens equation. Instead, practice drawing ray diagrams; and practice identifying the signs of each term in the relevant equations.

How I teach ray diagrams:

I only teach two principal rays for each optical instrument.  Why?  Simplicity.  Student difficulty in geometric optics does not come from the concepts themselves; the trouble is that all these lenses, mirrors, principal rays, objects, images, arrows, blend together.  

So for mirrors, I teach two rays only:  Parallel rays reflect to or from the focal point; rays toward the focal point reflect parallel to the principal axis.  For lenses, parallel rays either converge to or diverge from a focal point; rays through the center are unbent.

When the class practices drawing ray diagrams, I make them use these templates that I sketched out.  (Can you tell I flunked art class in seventh grade?)  The point is twofold: the templates force the students to use a full page for each diagram rather than cramming the diagram into a corner; and the rough sketch of the curved mirror saves the students from wasting all sorts of time using compasses or large circular objects to draw the mirror.*

* Seriously, before I used the template, a single mirror diagram could take a perfectionist student half an hour to complete, because he would obsess over the precision of the circular mirror shape. 

We identify by measurement the five relevant variables on each ray diagram: image distance, object distance, focal length, image height, and object height.  I've pre-drawn a table in the corner listing these variables in a column begging to be filled in.  The students get used to measuring di, do, and f along the principal axis, starting from the mirror or lens. 

Here, the trick is to require the students to include a sign for each entry.  What students think of as their failure to solve the thin lens equation correctly is almost never a true "math error:" rather, they generally screw up the signs of the terms in the equation.  So instead of practicing the calculation, we practice identifying signs.  

To simplify the conventions:

* converging instruments have positive focal length; diverging instruments have negative focal length.
* real images have positive image distance; virtual images have negative image distance.
* all object distances are positive [at this level].
* object and image heights are positive above the principal axis, negative below the principal axis.

I emphasize that for this exercise, a value without a sign is automatically marked wrong.  If a quantity is positive, I must see a + sign.  Otherwise, it's way too easy for a student to get most of the available credit by just putting numbers in the table; "oh, you know I meant positive, give me credit" is a disingenuous complaint when the whole point of the exercise is to force students NOT to assume that every measurement should be plugged in to the equation as positive.

It's worth practicing these diagrams for all four optical instruments until the students are bored with them.  Then optics questions on an AP or final exam will be thought of as free points.  


2 comments:

  1. Greg, I like how you simplify the rays for your students. Do you find that they spend much time concerning themselves with the minus signs? My AP students threw the guidelines in the trash last year and used their "intuition" to determine the sign of a magnification, distances, etc. They did alright with that. Thanks for the post.

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  2. Hi, Joe... Actually, they spend far less time concerning themselves with minus signs this way than when I used to have them make calculations; and, by using a few days when their ONLY mathematical job is to identify plus or minus on quantities, they get these sings right.

    It used to be, students would carry through calculations with the thin lens equation, the calculations would take as much time as the ray diagram (aargh), and the answers would all be wrong because they used the wrong signs. So they thought their problem was their ability to use fractions, rather than that they just couldn't remember sign conventions.

    Since I started doing things this way, I've gotten more right answers with less time expended by students. They've successfully separated "remember the sign conventions" from "solve the thin lens equation algebraically and plug in values."

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