# Inverse Square Law

#### gafftaper

##### Senior Team
Senior Team
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==Overview==
The Inverse Square Law states that the power of such things like electromagnetic radiation, acoustic energy, gravity, etc. is inversely proportional to the distance squared from the source of the radiation.
==In Lighting==
As light leaves a point source it travels in linear waves that are, in effect, perpendicular to the source. As the light travels away from the source it becomes increasingly less bright. Inverse Square Law tells us that the intensity of light produced, drops inversely proportional to the distance squared.
So: Final Intensity in lumens= Initial Lumen Intensity/(Distance from Source) squared. L=L1/(D^2), where L=Lumens on stage, L1=Initial output, D=Distance

If lamp produces 16 lumens of light at 1 foot, at 2 feet it produces 4 lumens, at 4 feet it produces 1 lumen... by 10 feet it produces only 0.16 lumens.

This is a real problem if your stage is 100 feet from your lighting position. lighting instruments are designed to produce extremely high initial light output using reflection (mirrors), refraction (lenses), and really high lumen output lamps.

Instruments with tighter beam angles produce a higher lumen output than instruments with wide beam angles. This is due to the higher concentration and uniform directionality of the beam of light. Thus, if you hung a 50˚ fixture next to a 5˚ fixture and both are the same distance from the stage, are the same fixture type (e.g., both are source fours), and have the same lamp in them, the 5˚ will be brighter.
==In Acoustics==
This same inverse square principle holds true for acoustic energy whether it be emitted by an instrument, speaker, person, etc. The [[Sound Pressure Level]], measured in [[Decibels]] (dB-SPL), at the point of measurement is equal to the original SPL level divided by the square of the distance to the source. Thus SPL=SPL1/(D^2).

Sound can also behave as an inverse proportion, so someone might want to check my accuracy here.

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#### David Ashton

##### Well-Known Member
Sorry guys this information is wrong when applied to theatre lights, inverse square law applies to a point source.
As you concentrate the beam and direct it nothing like inverse square applies, the extreme example is a searchlight shining miles into the sky, it only loses light by interaction with atmospheric dust vapour etc.
A look at any photometric data will show a similar story.
And the same applies to sound, affix a horn to your speaker and it will travel a long distance in a narrow band.
In brief Inverse Square has very few uses in theatre lighting and sound.

#### derekleffew

##### Resident Curmudgeon
Senior Team
I must disagree with you there, allthings. Please download my Excel spreadsheet here. Enter 0.1 for horizontal distance and whatever numbers you like for vertical distance. As you double the throw distance, the intensity is fourthed. Use it for PARcans, or Fresnels, or PARnels, the results are the same.

Thumbnail below is what I got for an ETC SourceFour 5° ERS. ETC's cut sheet on the fixture says: "To determine center beam illumination in footcandles at any throw distance, divide
candela by the throw distance squared." Candela is given for this fixture as 1,345,250. My numbers below are slightly off because ETC used to claim this fixture's candlepower as 1,370,000. (Coming soon, Conv_Photometrics_v5.xls, once I have time to change all of ETC's data).

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#### David Ashton

##### Well-Known Member
Right let us take a fresnel,it has a lumen output of "x", spot it down and you will get 700 lux at a point on the stage, flood it and you get 70 lux at the same point on the stage, obviously much larger area, but the same lumen output gives wildly different intensity at the same distance.
The lumen output of the lamp is the same for both states.
Where is the flaw in that argument?
Cyc light- very little variation at 1m 2m 3m.

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#### derekleffew

##### Resident Curmudgeon
Senior Team
Right let us take a fresnel,it has a lumen output of "x", spot it down and you will get 700 lux at a point on the stage, flood it and you get 70 lux at the same point on the stage, obviously much larger area, but the same lumen output gives wildly different intensity at the same distance.
The lumen output of the lamp is the same for both states.
Where is the flaw in that argument?
Your example relates to beam distribution, not the inverse square law. You didn't look at my photometric calculations, did you?

Let us say you want to make a pepperoni pizza, and have 113 grams of thin-sliced pepperoni. Now you could stack the slices on top of one another in the center of the pizza (spot focus) height about 7.6cm, or you could spread the slices evenly across the entire pizza (flood focus) height about 0.6cm.
The amount of pepperoni is the same in either case, but the heights are wildly different.
There is the flaw in your argument.

The lux output from a fresnel, or any other light source, will drop by a factor of 4 (2^2) when distance is doubled, provided the optics are not altered.

#### icewolf08

##### CBMod
CB Mods
Sorry guys this information is wrong when applied to theatre lights, inverse square law applies to a point source.
As you concentrate the beam and direct it nothing like inverse square applies, the extreme example is a searchlight shining miles into the sky, it only loses light by interaction with atmospheric dust vapour etc.
A look at any photometric data will show a similar story.
And the same applies to sound, affix a horn to your speaker and it will travel a long distance in a narrow band.
In brief Inverse Square has very few uses in theatre lighting and sound.
I have to agree with Derek on this one. What is the difference between a "theatre light" and a non theatre light. If you take a PAR38 that you bought at the hardware store and put it in your floodlight fixture in your back yard it is going to perform the same as a PAR38 you bought from your theatre supplier and put in a PAR can. The principles are the same.

The intensity of a light is vastly different if you measure it at different distances from the source not matter how the light is focused. If you take a measurement at 1m from the lens you will have more lumens than if you measure at 10m from the lens.

The same goes for sound. Sound is louder the closer you are to the source. If you take an SPL meter and take a reading 1m from the source it will be higher than a reading taken 10m from the source. This is why we often need fill speakers in large auditoriums.

Both sound and light spread out in a predictable fashion from a lamp or speaker respectively. If you have a fixture that focuses light, or a speaker that focuses sound, it will still spread out in a given pattern. Derek's pizza metaphor is pretty good. If you have a source that emits X lumens or decibels in a 20˚ field you have to cover significantly more area at 10m than at 1m. If the initial output is constant than less energy reaches every square meter at the further distance because it is less concentrated.

#### gafftaper

##### Senior Team
Senior Team
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Yes by a combination of reflector and lenses we can capture a point source and aim all of it's light in one direction, in parallel lines, defeating the inverse square law. But that isn't useful on a stage. If you take an older 6x9 ERS and remove the outer lens you will have an instrument that collects all the light and sends it out the front as close to parallel as possible. So that (in a perfect world) at 1' away it produces a 6" circle of light and at 100' away it produces a 6" circle of light and intensity will be the same at both locations (In reality that isn't quite true due to imperfections in optics and mirrors, but it's pretty close). This is also how your search light example works and it is great for search lights, however it's useless to us on stage. When was the last time you did a stage wash with 6" areas of light? No we need larger areas so, we put that second lens on the instrument. The second lens takes the controlled parallel beam, and focuses it to a point again and spreads it out gradually at a controlled rate. This is great because we are able to create the exact size pool of light we want, but unfortunately it means that the inverse square law also kicks back into effect again, because the beams of light are no longer parallel, in fact the lens forces them to a point source again out in front of the instrument. Which as you said the inverse square law effects point sources. Note that new fixed beam ERS's sometimes do all this with only one lens.

Trying not to be rude here, but you should be able to find this information in any lighting design book. I just got the classic 1958 edition of Stanley McCanless' "Method" as a Christmas gift. I refer you to page 33 on "intensity" for a description of how the inverse square law effects stage lighting. Or try one of the new lighting design books often recommended here from either Shelly or Gillette.

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#### David Ashton

##### Well-Known Member
I don't argue that a given stage light with a given setting and no parallel rays will observe the inverse square law, however zoom profiles and fresnels are designed to be adjusted and cheat the law.So any practical use of the inverse square law needs so many input parameters to be fed into it to be practically useless.
Alex said,"The intensity of a light is vastly different if you measure it at different distances from the source not matter how the light is focused. If you take a measurement at 1m from the lens you will have more lumens than if you measure at 10m from the lens.", true of course if you don't adjust the focus, totally untrue if you adjust your fresnel from spot to wide, my point which everyone is desperate to mis-interprate and deprecate is that theatre lights are designed to avoid this law to optimize light usefulness
If I am designing a factory lighting plan, I put in where the lights are and there position and can tell you the intensity at any point, in a theatre plan I would need so many parameters to give that information as to be totally impractical.
Which is my original point, in theatre there is little practical use for this law.
Derek,"The lux output from a fresnel, or any other light source, will drop by a factor of 4 (2^2) when distance is doubled, provided the optics are not altered."Which is what a theatre light is totally designed for, to be altered.
Q.E.D.

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Senior Team

#### David Ashton

##### Well-Known Member
Derek are you being deliberately obtuse? if you don't adjust the focus then the law applies, but theatre lights are designed to be adjusted, and let's apply the "law" to sound, put white noise through a speaker in your theatre and measure sound levels at 1,2,10 20m, there will be no useful correlation, the room acoustics make inverse square law of no practical benefit, the danger is that students see the phase "law" and think it is immutable, but of course the law is true in an anechoic chamber, but of strictly limited use in a real building.

#### amir

##### Member
This may help. Any light will follow this law UNLESS you change the way it's light is being modified. So you set any light up and the light coming from that light will follow the inverse square law.