The Inverse Square Law and its Implications on Cinematography

The Inverse Square Law and its Implications on Cinematography

Everybody knows the inverse square law - light gets dimmer by the square as it gets further away from the source - big deal, right? 

And yeah, you can compensate by dimming or boosting your lights. For example you got two actors talking to each other, you can ether light them with a 100 watt bulb from 1 meter distance, or with a 1000 watt bulb from 8 meter distance, the image is the same ...

... or ist it?

Wll, let's do the math.

First of all, the two lamps and the distance to the people:

A 100-watt light bulb produces about 1400 lumens.

A 1000-watt light bulb produces about 90,000 lumens.

The ratio of light output is therefore 90,000 / 1400 = 64.29.

According to the inverse square law , to achieve the same brightness on the person, the 1000-watt lamp must therefore be √64.29 times further away

than the 100-watt lamp, right

√64,29 ≈ 8,02

This means that if the 100-watt lamp is right next to the person (let's say 1 meter away), the 1000-watt lamp would have to be about 8.02 meters away to produce the same brightness on the person.

How does the light now affect our two actors. 

One scene is lit at 100 watts from a distance of one meter

The other scene is lit at  1000 watts from a distance of 8.2 meters

Even if you don't touch the shutter, iris or ISO of your camera, they are equally bright - I've just calculated that for you.

... but does the image therefore really look the same?

No, it doesn't  look the same. They look totally different.

Not only that, the shadows, the wrap, the background, the contrasts - everything changes when I move the light source closer or further away (all with the same brightness on the actors, and without adjusting anything on the camera).

What happened there?

Scenario 1: 100-watt light source nearby

Person A is standing right next to the light source - let's say 1 meter away

Person B stands 2 meters away from the light source

The light fall-off between person A and B is significant:

Person B receives only 1/4 of the light intensity of person A (2² = 4)

Scenario 2: 1000 watt light source at a distance of 8.2 meters

Person A stands 8.2 meters away from the light source

Person B stands 9.2 meters away from the light source

The light fall-off between person A and B is lower here:

Person B receives about 79% of the light intensity of person A (8.2² / 9.2² ≈ 0.79)

As long as I'm only shooting one person and they're not moving, I can manage with weak light sources that are close by if necessary (although the lighting effect isn't really nice, harsh and contrasty - but you can counter this with something else).

But the moment we have several people and they are moving around the room, that's the end of it. You then need more powerful lamps that you can place further away.

With low wattage close-up lamps, the light falls off way too fast, if someone moves just one meter, and they practically run into the darkness.

In addition, the whole square law calculation is based on an isotropic point light.

With an LED cob light this still works halfway decent, but with a Fresnel or even a reflector-lens light, which creates a “virtual light” far behind the actual light (e.g. Dedolights) this is no longer true, and panel LEDs behave differently again. A parallel attachment or a softbox also throw everything out of kilter, and you then have to make completely different calculations. 

Reflectors such as an UltraBounce comply with the square law to some extent, but the figures are completely different - the light decreases much less than you might think.

It's complicated -  much more complicated than many people assume.

And all that doesn't even involve a camera, ISO, aperture etc..

There are also lots of variables, if you take low light and camera ISO into consideration. 

Most folks think of ISO as an "in camera light switch", but it's actually more like a volume knob.  

Using ISO to compensate for a light source, that you want far away for aesthetics (like softer shadows and less harsh contrast) but is actually too weak, ins't always the best idea. 

It makes a huge difference which sensor I have, whether the camera is ISO invariant, has dual ISO, whether I shoot raw or not, digital or analog gain, bit depth, how I approach it in post, and whatnot.

In the Blackmagic Design camera family alone, there are three completely different concepts at work, so you can't even compare the Pocket cameras (Sony sensor) with the Ursas (Fairchild sensor) and the 12k/Cine Ursas (Blackmagic's own sensor). all three behave completely differently - of course, this also applies to all other brands.

Again: it's complicated, and - as always -  it depends.

There is no perfect camera and no "magic bullet" lighting scheme  - as a cinematographer you must know the limitations of your tools, and you must learn how to work within those limitations.