I found this forum on Google. This may not be the right section so excuse me if so. I have a rather simple question though. When you take a magnifying glass on a sunny day and position it just right over a piece of paper, the paper will start to burn. Is the inverse square law (distance) the biggest factor at play in getting the paper to burn? Or is it more that due to the way the lens is positioned, it concentrates the light into a single point rather than diffuse it?
No. Yes. The simple form of the inverse square law describes what happens to light propagating freely away from its source and dispersing as it goes. When you add a lens into the picture, it no longer applies.
On a camera, the focal length of a lens (how close/far the lens is from the film) has something to do with the amount of light that hits the frame of film. Is that because of the inverse square law?
No. The inverse square law is about how energy spreads out from an energy source. The focal length of a lens has more to do with sharp focus than with the amount of light on the film. It tells how far the film should be from the lens for the light from a single point of the object being photographed to be focused on a single point of the film.
Burning or not will depend on several factors. The area of the lens will determine how much solar radiation is captured (based on 1kW per meter squared - which is determined by the ISL but is the same for all lenses in clear sunlight). The temperature of the hot spot will depend on its size (the optics) and the rate at which heat is conducted away from the spot. A thin piece of paper, protected from drafts will burn easier than a fat piece of cardboard (or metal).
Adding to what others have said, the inverse square law would have a role to play if you were trying to burn the paper using e.g., a naked lighbulb. Since we can approximately treat the bulb as a point source radiating in all directions, the farther you get from it the less energy you catch. When treated as one, a lens is a completely different kind of source. It most definitely does not radiate equally in all directions, so inverse square law does not apply. There are other kinds of light sources that do not obey the inverse square relationship. Think of a prefectly collimated laser beam - no matter how far you get from the source, you get the same amount of energy per area. In principle, the inverse square law does play a role when trying to burn something with a lens in the sense that as you position the lens closer to the Sun, it catches a bit more of its energy than if you put farther away(i.e., closer to the ground). That's because the Sun does radiate (approximatelly)equally in all directions, so the amount of incident energy does fall as a square of the distance. Of course, since the change of distance of a few centimetres, or even thousands of kilometres, is so minuscle as compared to the distance from the Sun, the difference in received light is absolutely and completely negligible(so it's perfectly safe to assume the energy received from the Sun on Earth is independent of distance, and all light rays coming from it are parallel).
All else being equal (equal sized lens, equal distance from light source, equal distance from lens to film.), the focal length of the lens does not change the amount of light that hits the film as long as the film is large enough to catch all the light and nothing else blocks it. Focal length only changes how the light is focused.
This is not actually true. A source does not need to be isotropic for the ISL to apply. The power flux in a given direction does not affect the flux in another direction. All that is necessary is for the source to be small enough to regard as a point source. An extended source will exhibit the ISL once you get far enough away to regard it as a point source. (i.e. subtends a 'sufficiently' small angle).