Inverse Square Law & Reciprocity Law Failure

Inverse square law means that when the distance from the light source to an object is increased, the illumination of the object is inversely proportional to the square of the distance from the source. For me it’s easier to understand a law by writing a formula:
illumination=1/distance2
For example, if an object is placed 1m from a light source and then moved to 4m from the light, the light it will receive will bee 1/42=1/16 of the illumination.

At the same time opening the aperture (+1 f-stop) doubles the amount of light, accordingly closing (-1 f-stop) halves it. So to correct the exposure after the object was moved from 1m to 4m distance to the light source, we just need to decrease the aperture by 4 f-stops. For increasing distance I would write a formula in this way:
distance2=2(number of f-stops)

For example

  • if I double the distance: 22=2(number of f-stops) => 4=2(number of f-stops) => number of f-stops=2 (decrease aperture by 2 full stops to correct the exposure)
  • if I quadruple the distance: 42=2(number of f-stops) => 16=2(number of f-stops) => number of f-stops=4 (decrease aperture by 4 full stops to correct the exposure)

Shutter speed and/or ISO can be used to control the exposure as well. For example, if you need to add the amount of light equal to 3 full stops, you can open the aperture for 1 f-stop, decrease the shutter speed equal to 1 f-stop and increase ISO equal to 1 f-stop.

Here is detailed article on the subject: Rules for Perfect Lighting: Understanding The Inverse-Square Law

Below are test shoots concerning the subject.

IMG_6693web

f/8, 1/125 sec., ISO 400

IMG_6694web

f/4, 1/125 sec., ISO 400

The settings for the red cup (first image) were f/8, 1/125 sec., ISO 400. The distance to the cat (second image) was twice as big. That means we had to decrease the aperture by 2 f-stops to compensate the exposure while keeping all the other settings the same. So we get f/4, 1/125 sec., ISO 400 for the second image.

Next we practiced with the ambient light. In this case, our light source was a window.

IMG_6720web

f/4, 1/125 sec., ISO 400

f/2, 1/125 sec., ISO 400

f/2, 1/125 sec., ISO 400

f/4 for the first image, double distance for the second, that means -2 f-stops, so f/2 for the second image. We didn’t measure the distance and the sun is unstable light source that can change illumination in seconds, so the difference in exposure can be seen. It leads me to the conclusion that inverse square law is mostly usable when it comes to a constant source of light or a flash. It’s especially helpful to know the law when you don’t have a lightmeter or don’t have a lot of time to take frequent readings.

However, there are some cases when the law doesn’t work. It can happen in extremely low or high light conditions and is called Reciprocity Law Failure.  Most film manufacturers make correction data sheets normally available on their home pages. They usually come in the form of a spreadsheet or  a graph. To read the graph you just need to  find your indicated exposure time on the appropriate axis, look for the point where the curve has equal value and find a corresponding point on the corrected time axis. More on the subject here: WHAT IS RECIPROCITY FAILURE AND HOW TO CORRECT IT.

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