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Inverse Square Law

Published : 16th January 2007

Can anyone explain in simple terms the Inverse Square Law and its real world applications?

Thanks!

Liz Hinlein


Luminance, irradiance, or flux density produced by a POINT source is inversely proportional to the square of the distance from the source.

Light originates from a point, if the distance from that source is doubled, the area illuminated by the same amount of light from the source is 2 squared, or 4x the original area.

This also can pertain to sound...for twice the distance the amplitude is reduced to one-quarter

Exciting stuff, huh?

JR Anderson


The Inverse Square Law... yes... it's about calculating how bright a light will be. That's the real world application. By the way it's also know as the Law of Squares.

Let's say you want to put a light on a rooftop across the street from where you are shooting to simulate moonlight.... moonlight comes from above, right?... so you want the light high up. You can't afford a lift that can raise the light high in the air, so you tell your lowly electricians (that's me) to haul the big light up the interior stairs of the building and "just bloody well get it up there". The electricians might all quit because of your nasty way of saying it but they are all poor schmucks who are afraid of big Directors of Photography like you...so... they just do as they are told.

Now, you are not sure what F-stop to shoot at... should you use really fast fast film (which has lots of "bad" film graininess) or the slower stuff that looks better? Hmmm.... well, it would help to know how much light will be falling on your subject... that way you can figure out what f-stop you would be shooting. If you figure out that you are shooting a F 0.7... well, that limits your choices for lenses big time (very few are made to work at F 0.7). So then you would think to yourself... that light I thought would work over on that roof... she is going to have to be bigger! Much bigger!

But back to the inverse square law... if you know (from reading the manufacturer's specs) that a 10,000Watt light will give you X foot candles at a certain distance (foot candles are a measurement of light which says that a "standard candle" will put 1 footcandle of light on an object 1 foot away) you can then figure out how many foot candles will be at twice that distance.

How? Using the inverse square law of course!

The law says that as the distance of the light FROM THE SUBJECT increases... the amount of light reaching the same subject will decrease in proportion to the square of the distance from the subject. That's an annoying way of saying that if you know a 20,000 watt light will give you 200 foot candles at 75 feet (you know this from the manufacturer's specifications which you looked up online) then at 150 feet (which is the distance from the subject to the light across the street on top of
the roof) the light will be twice as far and...

150 feet is twice 75 feet... so that's 2x as far... the amount of light will be the square of 2... which is 4...the number of foot candles reaching our subject will only be 1/4 of 200....or... the amount of light reaching our subject will be 1/4 of 200 foot candles which is 50 foot candles.

Now to figure out how much that all is in F-stops... F-stops after all are the units of measurement that are written on the lenses and are generally used in those fancy light meters all the Directors of Photography have these days. (If you don't know what an F-stop is...well, I'm not going to get into that here... this would take all week!)

We know from... well, who knows how we know it... but somebody knows that if you have 100 foot candles of light hitting a subject and you are using 100ASA film... your light meter will tell you that you will need to shoot at an F 2.8 to expose your subject properly. So if, in our real life shooting situation, we only have 50 foot candles coming from our big 20,000 watt light across the street... then the light meter will tell us we have to shoot at a F 2.0.

Tada! We now know we will be okay to shoot with 100 ASA film... F 2.0 is plenty of light and we are good to go.... lots of lenses are made to shoot at a F 2.0, after all.

PS - I have completely made up the number for how much light a 20,000 watt light will deliver 150 feet away... you will not be able to use 100 ASA film at night with only a 20,000 watt light on the rooftop across the street, let me assure you of that. Okay..... maybe you could, but you'd be crazy! But that was not the point... the point was that to learn about the inverse square law. I hope this helped.

Piotr Jagninski
Gaffer - New York City


For a point source of light, if you double the distance (x2) you get a quarter (x1/4) the light (luminance). That is, 2 stops less.

If you have three times the distance, the light drops to 1/9th. In general, if you increase the distance by a factor of X, the luminance changes by 1/(XxX) - that's 1 over X squared.

This is only good for sources of energy (light, sound, heat etc) that are radiating outwards from a point. So it's not good for spotlights, or for large area softlights etc. It works for the sun, but you have to travel a long way to double its distance. So an artificial light needs to be as distant as possible if you want it to emulate sunlight.

Also works for sound (hence the need for mikes to get in close to an actor - their voice falls off in proportion to the square of the distance to the mike, while ambient noise remains constant.

And it's true for gravity and radiation too.

Dominic Case
Atlab Australia


For non point source like a Chimera etc use this link to a calculator :

http://www.lighttools.com/eggcalc/index.html

About EggCalc

Predicting the illuminance the amount of light at some distance from a large diffuser can be difficult. The inverse square law only applies to point sources of light, which hardly describes a diffuser when viewed at close range!

There is an equation that perfectly describes the change in illumination with decreasing distance, but it is unfortunately far too cumbersome for everyday use.

The problem becomes more acute when you place an egg crate screen (such as a Lighttools Soft Egg Crate) in front of the diffuser to control the light distribution. Predicting the beam spread with any accuracy involves some very cumbersome calculations just the job for your personal computer!

EggCalc performs these calculations for you. It determines the distribution of light from large diffusers at any distance, both with and without egg crate screens.

Ed Barger
BARGER-BAGLITE


Liz Hinlein writes:

class="style15">>> Can anyone explain in simple terms the Inverse Square Law and its >>real world applications?

OK. In simple, real-world terms, the intensity of light emanating in all directions from a point source (i.e., a clear light bulb with no reflector) diminishes with the square of the distance. In other words, when you double the distance, you divide the intensity by four.

In filmmaking, however, you're rarely using bare sources with no reflectors. When the light is no longer emanating equally in all directions but is being concentrated, diffused, etc., things get complicated. So instead of calculating light intensity we usually just measure it -- it's more practical and usually more accurate. Formulas have their place, but they're used mainly in specialized situations, or as starting points.

I hope this helps.

Dan Drasin
Producer/DP
Marin County, CA



>>So instead of calculating light intensity we usually just measure it

Yes, good for final accuracy . . but surely knowing that if you move a key light twice as far away from the subject (for whatever reason) you are going to lose not one stop but two stops is useful before you put the light there. And having an idea of how far down the street your truckload of lights will reach - and therefore how many trucks to bring;-) - is useful before you arrive on set with a couple of 2Ks.

Dominic Case
Atlab Australia


One of the more practical ways to think about the inverse square law was something I read here on CML, probably in this very forum. It was so simple I was kicking myself for not thinking of it sooner. Much sooner. In a past life, even.

Take the standard F-stops (.7, 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22,
32, 44, 64, etc.) and think of them as distances. As examples,

From 4' to 5.6' you lose one stop.

From 4' to 8' you lose two stops.

From 16' to 22' you lose one stop.

From 16' to 32' you lose two stops.

Somewhat brilliant, painfully obvious, and very handy. And it's all about the inverse square law.

Art Adams
Director of Photography
Film | HiDef | Video
Mountain View, CA, USA
www.artadams.net


Art Adams writes:

class="style15">>>Take the standard F-stops (.7, 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 44, >>64, etc.) and think of them as distances. As examples,
>> From 4' to 5.6' you lose one stop.
>> From 4' to 8' you lose two stops... etc.

If you mean this other than simply as another expression of the inverse proportion, it could be confusing in real-world terms, IMHO.

The only light that actually drops off with the square of the distance would be from a bare bulb floating in empty space.

If you're talking about increasing the distance of the camera from the light source or from what it's illuminating, that doesn't change the level (or exposure) at all. As you back the camera away, the total amount of light impinging on the film or sensor diminishes because the light source gets *smaller*... not dimmer. All else being equal, if your exposure is f/4 with the camera 10 feet way, it will still be f/4 when the camera is 100 feet away.

Another way of looking at this is that an incident light meter at the subject's position will correctly indicate the proper aperture setting, regardless of how far the camera is from the subject.

Dan Drasin
Producer/DP
Marin County, CA


class="style15">>> If you mean this other than simply as another expression of the >>inverse proportion, it could be confusing in real-world terms, IMHO.

The point that I was trying to make is that if you take the standard f-stops and convert them into distances the drop off works the same way :

From f8 to f11, exposure drops by 50%

From 8' to 11', the amount of light drops by 50% from a point source

Similar but different, no?

Art Adams
Director of Photography
Film | HiDef | Video
Mountain View, CA, USA
www.artadams.net


Art Adams writes:

class="style15">>>From f8 to f11, exposure drops by 50%
>>From 8' to 11', the amount of light drops by 50% from a point source

Ah, so. Very cool!

Thanks, Art. I've learned something!

Cheers...

Dan Drasin
Producer/DP
Marin County, CA


Dan Drasin writes :

class="style15">>> From f8 to f11, exposure drops by 50%
>> From 8' to 11', the amount of light drops by 50% from a point source

But that is probably the only instance where that works.

From f2 to f2.8 the exposure also drops by 50%, but I doubt the amount of light from 2' to 2.8' drops by 50%. (but I confess I haven't tested this!)

Doug Hart
1AC, NYC


>> From f2 to f2.8 the exposure also drops by 50%, but I doubt the >>amount of light from 2' to 2.8' drops by 50%. (but I confess I haven't >>tested this!)

The inverse square law is not an approximation, it's a fundamental law of physics - so it would work even at these distances - provided you had a _true_ point source. Close up, we'd be talking about a small filament with no reflectors or anything.

Dominic Case
Atlab Australia


Art Adams wrote :

>>One of the more practical ways to think about the inverse square law >>was something I read here on CML, probably in this very forum. It was >>so simple I was kicking myself for not thinking of it sooner. Much  >>sooner. In a past life, even.

Art, this is why I read CML. 

This is the best and easiest explication of the inverse square law I've ever seen.  And now that I am teaching (college students who seam to have no math skills beyond addition and subtraction these days) it gives me something that is easy to explain and I hope easy to understand.   

Still kicking myself,  

Steve Golden, ICG
DP Chicago


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