Teaching inverse square laws

by Stephanie Chasteen on February 1, 2012

That darn inverse square law comes up in so many places — electric fields drop off as 1/r^2, so does light intensity, gravity, and a bunch of other things that I’m not thinking of at the moment.

I just came across some fun things in my archives from some master teachers on how they teach the inverse square law.

First, here is a really nice example of expert use of screencasting to explain a nice single concept like this:

Deb’s inverse-square-law presentation at Menlo School.

by Paulmann on Wikimedia

I can imagine students watching this little video several times.  She gives a nice pictorial representation of how intensity drops off over distance, using a physical analogy (butter sprayed on toast).  She gives a formula.  She shows how the formula applies in a few instances.  And it’s nice because the handwritten form of this screencast (rather than PPT slides) forces her to slow down to thought-speed.

Craig Young, from Sacramento schools, shared this activity that he uses to help students “discover” the inverse square law on their own:

Materials: 1 sheet black construction paper, 1 sheet graph paper with 1 cm lines, 1 MagLite, 1 meter stick

1.  Set up a MagLite flashlight in candle mode (removing the lens over the bulb).  It works best with MagLites because they have a very small bulb – almost a point source.

2.  Cut a square 1 cm on each side out of the middle of a sheet of black construction paper.

3.  Put the MagLite at the end of the meter stick.  Hold the construction paper vertically at 10 cm.

4.  Start with the graph paper at 10 cm, on the opposite side of the construction paper from the MagLite.  Students count how many squares on the graph paper are illuminated (quantitative) and how bright they look (qualitative).  It should be one square at this initial position.

5.  Move the graph paper to the 20 cm mark and repeat the counting and brightness observation.

6.  Repeat step 5 moving the graph paper 10 cm further each time.  If you can turn out the lights in your room, students should be able to get data out to about 80 cm (8 data points).

The light spreads out, covering more squares at each consecutive distance.  For gravity and electromagnetic forces spreading out in a complete sphere, the area of the sphere increases with the square of the radius of the sphere.  So the intensity decreases as the inverse square of the radius

I’m always struck by how many of these high school level activities would be appropriate at the college level to make sure that students have an intuitive grasp of what some of these numbers really mean.

On the other hand, it’s easy to imagine light particles spraying out and falling off as the inverse square, but how does one help students understand that electric and gravitational fields do the same when there’s nothing “spraying” out, except for those sort-of-physical field lines?  That next level of abstraction is definitely a difficult one.

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