Let there be light!

In the beginning, there was light.

Imagine a very bright source of light which is perfectly monochromatic (just one wavelength.)

Imagine you are in the room next door with a pinhole looking at it. (like the movie Psycho.)

The number of photons streaming past you is proportional to the intensity of the light (how "bright" it is), the size of the pinhole, and the length of time that you look.

Let's assume the pinhole is circular. (This will not really matter, as you will see later.)

In physics, instead of measuring the number of photons of a monochromatic source, it's simpler to measure their total energy because they are proportional. (E = h × ν.)

Illuminance is basically the amount of energy that is pumped out per unit area of the source.

(Aside: it's a little more complicated for light that is not all of the same color but let's ignore this point just for the time-being. We will return to this point later.)

Let L be the illuminance. If the pinhole is perfectly round and has radius r, and you look for time t.

Luminous Energy ∝ L × r² × t


That symbol (∝) which stands for "proportional" refers to the fact that the two quantities are related with a hidden constant — in this case, π, if you want to get all technical.

This should be very very intuitive. Double the illuminance (energy) pumped out, and well, you get double the energy (DUH!)

Double the time, and twice as many photons stream past.

Increase the radius, and the area increases as the radius-squared. This is the only tricky part.

Memorize this stuff. Everything that follows will be a direct consequence of this formula.

That pesky little "square" of the radius will play a disproportinate role in what follows. Take a careful note of that little bastard!