# Inverse square law

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Posted 25 April 2014 - 07:55 AM

Do all lights(luminaires) follow inverse square law?

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### #2 Zac Fettig

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Posted 25 April 2014 - 08:23 AM

No, just unfocused lights. The inverse square law assumes a light source is a point where light is allowed to radiate uniformly in all directions.. If the light is focused (say by a parabolic reflector), the light will drop off, but not necessarily by the square of the distance. If it did, spot lights would have to be much more powerful.

Lasers, for example, frequently do not follow the law, since the beam is focused to the point that it doesn't expand. The beam is only subjected to scattering and atmospheric diffraction. It will get less intense but not necessarily by the square of the distance.

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Posted 25 April 2014 - 01:04 PM

Thanks a lot zac for the information.

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### #4 Sharath George Benny

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Posted 28 April 2014 - 11:27 AM

No, just unfocused lights. The inverse square law assumes a light source is a point where light is allowed to radiate uniformly in all directions.. If the light is focused (say by a parabolic reflector), the light will drop off, but not necessarily by the square of the distance. If it did, spot lights would have to be much more powerful.

Lasers, for example, frequently do not follow the law, since the beam is focused to the point that it doesn't expand. The beam is only subjected to scattering and atmospheric diffraction. It will get less intense but not necessarily by the square of the distance.

Doesn't it? Except for the case of laser you mentioned, which is an exception. Luminaries does follow the inverse square law. Even for focused lights. If you check the the photometric data of a light fixture (like PAR), you'll notice that the light intensity in lux at any point multiplied by the square of the distance to that point from the light source will give you the total output of that source in lux.

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### #5 Travis Gray

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Posted 28 April 2014 - 11:32 AM

Doesn't it? Except for the case of laser you mentioned, which is an exception. Luminaries does follow the inverse square law. Even for focused lights. If you check the the photometric data of a light fixture (like PAR), you'll notice that the light intensity in lux at any point multiplied by the square of the distance to that point from the light source will give you the total output of that source in lux.

Yeah. It's just with a focused light, you're concentrating it into one spot, so at the same distance as an unfocused light, it looks more intense because you're focusing more of the light into a spot. But it will still follow the same falloff rate as any other light. Which, that sounds super repetitive.

I won't comment on lasers. I don't get that stuff. It's the devil's technology.

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### #6 Zac Fettig

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Posted 28 April 2014 - 12:57 PM

Inverse square law always assumes light (or similar free wave energy) is a point source, free to radiate in all directions equally. When you focus it, with a lens, it's true you simply concentrate it. If it's a parabolic reflector, it also orients the light (assuming light is a point source at the focal point of the parabola). This is why a flashlight will throw light so far with a relatively weak bulb.

Here's another (hypothetical) example: Surround a light with a big sphere, coated matte black on the inside, with a single hole cut out. Add a long tube over the hole, also black coated on the inside. Two things will happen. The surrounding sphere will get hot (black body radiation), and the light that comes out of the tube will not drop off at the inverse of square of the distance. If the tube is sufficiently long, all light will be oriented to make it as straight as the laser beam example, but with the color spectrum of the original light source, not a single frequency.

Practically, it doesn't make a difference. The second you add diffusion over a light, it is no longer oriented, and the inverse square law applies.

http://en.wikipedia....erse-square_law

http://en.wikipedia.org/wiki/Parabola

http://en.wikipedia....bolic_reflector

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### #7 Sharath George Benny

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Posted 28 April 2014 - 01:36 PM

Inverse square law always assumes light (or similar free wave energy) is a point source, free to radiate in all directions equally. When you focus it, with a lens, it's true you simply concentrate it. If it's a parabolic reflector, it also orients the light (assuming light is a point source at the focal point of the parabola). This is why a flashlight will throw light so far with a relatively weak bulb.

Here's another (hypothetical) example: Surround a light with a big sphere, coated matte black on the inside, with a single hole cut out. Add a long tube over the hole, also black coated on the inside. Two things will happen. The surrounding sphere will get hot (black body radiation), and the light that comes out of the tube will not drop off at the inverse of square of the distance. If the tube is sufficiently long, all light will be oriented to make it as straight as the laser beam example, but with the color spectrum of the original light source, not a single frequency.

Practically, it doesn't make a difference. The second you add diffusion over a light, it is no longer oriented, and the inverse square law applies.

http://en.wikipedia....erse-square_law

http://en.wikipedia.org/wiki/Parabola

http://en.wikipedia....bolic_reflector

Im sorry Zac, but I don't understand what you're trying to say here. You earlier said that luminaries don't necessarily follow inverse square law. I do understand the inverse square law, parabola and the parabolic reflector. But I don't see why luminaries don't necessarily follow inverse square law as you mentioned in your first comment.

It'd be easier if you could say which light fixture in specific you have in your mind, that doesn't follow the law except for a laser?

Edited by Sharath George Benny, 28 April 2014 - 01:40 PM.

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### #8 Travis Gray

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Posted 28 April 2014 - 01:56 PM

Im sorry Zac, but I don't understand what you're trying to say here. You earlier said that luminaries don't necessarily follow inverse square law. I do understand the inverse square law, parabola and the parabolic reflector. But I don't see why luminaries don't necessarily follow inverse square law as you mentioned in your first comment.

It'd be easier if you could say which light fixture in specific you have in your mind, that doesn't follow the law except for a laser?

I kinda get what he's going for, like with a laser, since there's no spread, or very little spread (again, lasers, I don't know), but even with a focused light, say something like a Source 4, that light still spreads from the lens to the target. So it's still going to follow the inverse square law. It's just more intense closer to the light because of its focus, so, that means it'll be more intense than an unfocused light throughout the path.

So if you have 100 lux with a focused light 10ft out, it'll be 25 lux 20ft out.

If you have 50 llux with an unfocused light 10ft out, it'll be 12.5 lux 20ft out.

At least that's how I always understood it. And calculations always back that up.

Edited by Travis Gray, 28 April 2014 - 01:57 PM.

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### #9 Gregg MacPherson

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Posted 28 April 2014 - 05:27 PM

I did a really rough experiment on this recently. I knew that point sources had to follow the inverse square law, and at a distance lamps will behave like point sources, but I knew that working closer to lamps they don't. My experiment was really rough, but a redhead and a 5" fresnel both looked closer to linear than inverse square at 4'/8'. Only one stop variation. But I had no time, bleeding daylight and only guessed the measurements. And the lamp illumination wasn't very even.

So I did some basic math, which actually fits my expt result. If anyone want's to look PM me an email and I will send the spreadsheet. You can vary the reflector size and beam angle and look at the falloff over distance. Maybe my math is as rough as my expt.

If you do a drawing of a spread beam from a point source the inverse square law makes intuitive sense. If you draw a reflector with a spread beam it does not, easpecially for narrow beam angles.

Time for Guy Holt to step in and sort us all out.
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### #10 Chris Millar

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Posted 28 April 2014 - 07:09 PM

I think with the appropriate diagrams people will see what Zac is taking about. It's a snip too complex for words alone... Terms get confused etc.
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### #11 Zac Fettig

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Posted 29 April 2014 - 06:40 AM

Sorry about the confusion! I was just trying to help.

There are always a lot of caveats in physics, to account for the real world. Which is why so many textbook physics problems began with statements like: "Take a spherical cow of uniform density in a perfect vacuum..."

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### #12 John E Clark

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Posted 01 May 2014 - 06:06 PM

Im sorry Zac, but I don't understand what you're trying to say here. You earlier said that luminaries don't necessarily follow inverse square law. I do understand the inverse square law, parabola and the parabolic reflector. But I don't see why luminaries don't necessarily follow inverse square law as you mentioned in your first comment.

It'd be easier if you could say which light fixture in specific you have in your mind, that doesn't follow the law except for a laser?

The way the 'inverse square law' is stated, is in reference to a point source, and distance from said source.

Think of a light bulb, perhaps 2 inches in diameter and you are standing a 5 feet. At that distance, the light bulb diameter is 'small' relative to the distance to the observer, and so is a 'point source' and so the inverse square law applies.

With something like a soft box, where the diameter of the surfice is 'large', relative to the location of the subject, then the inverse square law does not apply... I'll leave it as an exercise to figure out at what distance the a 4 foot diameter soft box become essentialy a 'point source'...

In any case, one can perform a gedankenexperiment in that case and not that if one is 1 foot away from a 4 foot softbox, that by going to 2 feet... one does not '1/4' the light that one had at 1 foot...

I've seen some highschool science fair type experiment were someone has performed the metering at various distances until the readings begin to follow the inverse square law... I think it's about 2-3 x the diameter of the illuminating surface...

To understand a 'focused beam', via a 'Frensel lens', and note that the beam can be viewed as a point source, but the point source is not located where the lamp head is placed, but at some distance to the 'back' of the housing, say 200 feet... because the beam is diverging only a small amount due to the focusing lens. With the idea that the 'point source' location for the frensel lens lamp is 200 feet away, and the lamp directed to a subject at 10ft, and then moved to 20 feet, because the 'virtual point source' is 200 feet distant, I've only moved the subject 'virtually' from 210 feet to 220 feet from the virtual point source location... and so I've not 'doubled' the disance from the point source virtual location.

This also 'works' for a laser, where the 'point source' is infinitely far away... well... not really, but definitely must further than most things...

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### #13 Gregg MacPherson

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Posted 01 May 2014 - 07:37 PM

@jeclark2006

I'd like to do properly my sofar very rough experiment with a redhead and a 5" fresnel when I have time.  One problem I have is that the illumination is not even.

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### #14 John E Clark

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Posted 02 May 2014 - 11:44 AM

@jeclark2006

I'd like to do properly my sofar very rough experiment with a redhead and a 5" fresnel when I have time.  One problem I have is that the illumination is not even.

All lights have fall off from the center... just some lights are worse than  others... take your measurements from the beam center, and then set up  the light on an even surface, and see if you can characterize the fall off from beam center.

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### #15 John Miguel King

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Posted 15 May 2014 - 03:11 AM

Please allow me to enter the conversation.

The element missing in the dicussion is the concept of "flux". It is this, the flux of photons, that we measure. The flux of photons of, say, a fully spotted fresnell is much higher than that of an equally powerfull bare bulb at the same distance.

What the inverse square law defines is the how flux changes. As you'll see in the wiki page, one (if not THE) most important variable in the equation is the the angle of the photons as measured off the radiant surface.

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### #16 Rob McGreevy

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Posted 08 June 2014 - 05:53 PM

My understanding of this (and please correct me if I'm wrong) has always been that the inverse square law always applies, but it's also proportional to the spread of the beam from the source. So you have a beam of light, and the amount of light being emitted never changes. But the beam spreads, so at a certain distance the same amount of light is covering an area 4x as big, so it appears to be 4x less bright. It's the same amount of light, but it appears to be less because as it travels from the source and spreads out more it has to cover a broader and broader area, until the spread is so wide there appears to be no light at all. There are still photons traveling, there are always photons traveling. But once the spread gets wide enough you don't see them anymore because they are spread so thin. So the falloff is happening, but if the beam is very concentrated it will be a greater distance when it reaches the point where it has to cover 4x as much area and therefore cause that inverse output relationship we've all come to know and love. I've pointed 5 degree source fours directly at the sky, and on a cloudy night you can make a bat signal. It'll reach, but it's much dimmer -  that's how long it takes the beam on those to really spread an appreciable amount and cause noticable fall off.

As I understand it this principle even applies to lasers, it's just that lasers are so concentrated it takes forever for the beam to start to spread and thereby use the same amount of light to cover a wider area. But I've seen weaker lasers pointed at great distances, and you can see them start to spread and the ouput get weaker as that happens.

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