Re: Inverse square law

[Steve Hodges <shodges@xxxxxxxxxxxx>]

Hans Klemmer wrote:
> Jeff, 
> I have been lurking here for a while, and sending in comments now and 
> again. I'll scan some of my recent LF negs. and post them  when the 
> semester quiets down. The point to my previous message is that there 
> has been a lot of good information exchanged in this forum in the past 
> and I am guessing that most of don't want to read personal arguments. 
> Now, if you would like me to begin a discussion by asking a question, 
> I will; how do the rest of the instructors in this forum teach the 
> inverse square law to new students?
I've often done this using a waving of hands rather than actual paper as 
I've never had to explain it in front of a group of people.  I've seen 
it done on a blackboard in this way, but I find that there are some 
people who can't easily grasp the 3D example drawn on a blackboard, and 
trying to draw perspective images is likely to illustrate why you became 
a photographer rather than a painter.

Roll up a piece of paper into a cone and cut it off square.  Sit it on a 
table.  Discuss how a single point at the apex is the source of a cone 
of light.  At any point the circle you get as you cut across it 
represents the area over which the light is spread.

Measure the diameter at the wide end. Remember to divide by 2 to get the 
radius :-)  Show the area is pi*(r)^2 -- hopefully everyone will 
recognise that.

Cut the cone half way up.  Measure the diameter (show that it's half 
what it was at the open end)  Show that the area is now 1 quarter of 
what it was.

Discuss -- If the same amount of light is spread over 4 times the area 
when you double the distance, then how much light reaches the same size 
area at each of these 2 distances?  the ratio must be 4:1

Stop here!  Generalise and say that it's therefore an inverse square 
law.  You haven't proven it, but anyone capable of seeing you haven't is 
quite capable of going that one step further to show it must be.

If you make up the cone before class, make it as close to 22.6 cm in 
diameter at the base as you can (and cheat a little when you measure it 
so it *is* 22.6 cm).  This gives you close to 400 square cm of area, 
which makes some of the discussion a lot easier.  If you make up a more 
permanent example, you could make a series of square pyramids all with 
the same angle at the apex.  (place the smaller over the larger to 
demonstrate).  4 half height pyramids will neatly cover the area of the 
base of the full height pyramid, allowing you to show that the level of 
illumination is 4 times greater from half the distance without resorting 
to Pi, squares, or multiplication, concepts that may be foreign (gasp!) 
to some students.

Four shorter square pyramids can be held together to show that when the 
light source is indeed 1/2 the distance, the illumination for the other 
three quarters of the area is indeed coming from *different* cones of light.

You can also use a cone like this to demonstrate why light is dimmer 
when the sun is lower in the sky.  Proving the mathematical relationship 
between the angle and the ratio of areas illuminates is non-trivial.

This method glosses over a number of details, such as the fact that 
certain light sources you may actually use do not obey an inverse square 
relationship, but that is a whole other kettle of fish.


Steve