I know it's scary ...
I did stunningly bad at maths at school. I got through my O Levels (I'm pretty old) reasonably well but a long bout of school absence, caused by the removal of my appendix, caused me to miss most of the Trigonometry section and I took ages to catch up! The least said about my A Level Pure and Applied Mathematics the better! I did recover the situation somewhat when I went to college.
The trouble is, for photographers, there is a little bit of maths involved, f-numbers or stops and things like that. When it comes to lighting there is even more and one of the key ones is something called the Inverse Square Law - which I hasten to add relates to light in general.
Light Intensity = 1/Distance*Distance
What? Well, put simply it means that the further a light source is from the subject it is illuminating, the less intense it is. That may seem obvious but you might think that if you move a light source twice as far away it becomes half the intensity but that is not the case. This is where the 'Distance*Distance' bit comes in, the 'square' bit, and just how the distance effects the light can be seen by putting some real world values into the equation.
1/(1*1) = 1/1 = 1
In the above example our distance is set to 1m but now let us increase the distance to 2m
1/(2*2) = 1/4
What actually happens by doubling the distance from 1m to 2m is the light intensity is now only a quarter of what it was. Things get worse if we move the light out to 3m
1/(3*3) = 1/9
Now we only have nearly a tenth of the power we started out with. This is how it looks in diagram form.
What use is that to me?
On the face of it you might think that you do not really need to understand what is going on here. Trial and error would show you that if you moved the light source away from your model you would turn the power up to compensate. To an extent that is true but the distance a light is from a subject has some nuances. With respect to the inverse square law itself the key thing is what happens to our background.
We are going to dispense with the maths at this stage because it is more useful to see what the inverse square law does rather than try to work it out. We do not need to do the calculations. In the three shots below, featuring my new, very cheap model, Amy, I have moved the flash from 1m in the first shot, out to 2m in the second shot and finally 3m in the third. As I have moved the flash out I have increased the flash intensity to keep the exposure on Amy's face the same (more or less) so she is correctly exposed in all three shots. I have not changed anything else in my setup.
Sorry I need some more maths
The very important thing to notice here is what has happened to the background. The distance between Amy and the background is about 2m. The flash head is only 1m from Amy so she gets the full intensity of the flash using our inverse square law 1/(1*1), however the background is effectively 3m away from our flash so it only receives 1/9 of the intensity 1/(3*3). Consequently it appears darker. Put another way, Amy is receiving 9 times as much light as the background.
As I move the flash head further away from model the relative difference in distance from the flash to model and the flash to the background becomes less, consequently the drop off in light between the model and the background becomes less. The distance from the flash to the model is 2m and I turn up my flash to compensate for the drop in intensity (2 stops). The model is receiving 1/(2*2) i.e. 1/4 of the flash intensity. Our distance to the the backdrop is now 4m so the loss in intensity is 1/(4*4) i.e. 16th of the power. Again, to put it another way, the model is get 4 times as much light as the background.
Last leg, with the flash 3m away from the model, and 5m from the background, the background is only receiving 1/(5*5) = 1/25th of the flash intensity and the model is receiving 1(3*3) = 1/9th of the intensity. Again, flipping this on its head the model is getting just shy of 3 times as much light. I have increased the flash intensity by another 1 stop to compensate for moving it back a further 1m.
I don't follow, what should I take away from this?
The further we pull the flash out the brighter the background becomes in relation to our model because there is less light fall-off. If you want to keep your background dark place the light source as close the model as possible.
Didn't you just create the background in Photoshop?
No, the examples above are done using proper light modelling software that takes into account light source power, distances, reflective surfaces etc. Below are the full lighting diagrams for each of the distance.
One of the other side effects of the light distance is how hard or soft the light appears. By this I mean how strongly it casts shadows. In general the closer the light source is to a model the larger its effective size is. Consider the sun, it is very large but also a long way away so actually becomes a relatively small light source and we see this by the harsh shadows it causes on bright days. On a cloudy day the clouds diffuse the sunlight making them a much larger and softer light source casting little if any shadows.