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Bow to Your Partner for Some Inverse Square Dancing

7/8/2013Jay Holben |

When the venerable
Bill Klages
last left the
pages of *TV Technology,*
he was discussing
some mathematical esoterica
regarding inverse
square law and I was a
bit inspired to pick up,
somewhat, from where
he left off.

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We all know inverse square law as the rule of physics regarding point sources where the intensity of the source will diminish by the square of the distance traveled. For most lighting professionals, this is a rule to determine fall off of a fixture at a specific distance.

**YOU DO THE MATH**

This works two ways. If you have a fixed
relationship between your fixture and your
subject, then using inverse square can aid
in determining the size of fixture necessary
to light your subject to a specific stop.
The opposite of this is if you have a fixture
that is providing too much
intensity, knowing inverse
square can help you quickly
determine how much light
you can reduce by simply increasing
the distance to the
subject to obtain the intensity
you’re looking for.

For the mathematically inclined:

Intensity = 1 divided by the square of the distance.

Looking at some photometrics for lamps, a typical 1K open-face fixture has an intensity of 1,000 footcandles of light at 4 feet. If we increase that distance to 8 feet, our intensity drops to 250 footcandles. We have doubled our distance, but our intensity is now 1/4 (or the inverse square the distance traveled) the output.

If we look at Fig. 1, which is typical to illustrate inverse square, we see that as light rays diverge from a point source, they travel outward in all directions and the further away from the source the more they diverge and cover a larger physical area. So at 1 foot, what might cover a 1-foot square area, at 2 feet will now cover a 4-foot square area. At 3 feet, that same light is now spread out over a 9-foot square area. If we continued to 4 feet, we’d see the light spreading over 16 square feet of area. So from 1 foot to 4 feet, the light has diminished in intensity 1/16 of its initial power because it is spread over that much more area.

**ANIMATED SUBJECTS**

When you’re dealing with a fixed subject,
say a news anchor sitting at a desk or
a seated interview subject, the relationship
of that subject to the light source doesn’t
change. If, however, that interview subject
is very animated and leans forward in their chair, their relationship to the light source
can change substantially. This is where understanding
inverse square can help us alleviate
problems before they arise.

If we’re lighting that interviewee with a 4-foot softbox from four feet away and the subject leans forward during the interview, moving one foot closer to the fixture, that subject has changed their relationship to that light by 25 percent—or the reverse of inverse square there’s about a 5 percent increase in intensity on that subject; definitely a noticeable brightness.

This is a situation where bigger is better. The 4-foot soft source at four feet has the same quality of light as the 8-foot source eight feet away, but the spatial relationship to the subject is substantially different.

This also works for lighting groups of people— the further away and larger the source is, the more evenly the light will spread among the people. From one to two feet away from a fixture there’s a drop off of 75 percent light intensity, but from that same source, the distances from eight to 12 feet (room for three people to stand) there’s a drop off of less than 1 percent among them.

If we have the ability to increase that source to a 20-foot source 20 feet away, then the intensity difference between our subjects is absolutely negligible and they can move within a substantial area without changing light intensity and exposure.

The practical lesson here is that the larger and further away the source is, the larger an area we can cover with relatively equal illumination from a single source. This is what I like to call the lighting inverse square dance!

*Jay Holben is the technical editor of
*Digital Video* and a contributor to *Government
Video*. He is also the author of
the book, “A Shot in the Dark: A Creative
DIY Guide to Digital Video Lighting on
(Almost) No Budget.”*