"TV Technology"
01/02/2013
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TV Technology
Television Lighting: Inverse Square Law
by Bill Klages
The Law
The Inverse Square Law of Illumination has been with us for so long and is so commonly taken for granted that it might seem a questionable subject for discussion. However, I feel a revival of interest in this deep-rooted rule is justifiable – as a result of surprising results observed in a recent set of tests I conducted.
The Basics
The Reverse Square Law clearly states that it is only applicable in the case of a point source of light. We usually don’t use point sources directly, only as part of an instrument. Typical practical point sources of a small size would be short-arc discharge lamps as well as compact tungsten filaments. A corollary that is always added to the Reverse Square Law is that, for the purpose of estimating Illuminance (foot-candles or lux), a source can be assumed to be a point source if the distance from the source to the illumination plane is at least five times the widest dimension of the source. Some more conservative illumination experts say that this distance should be greater than ten times. For example, to be able to use the Law for a 6-inch Fresnel spotlight – and taking the more cautious approach – the distance from the fixture to the light meter must be greater than 30 inches. Therefore, if I measure 600 foot-candles at 5 feet from my spotlight, the Inverse Square Law says I should measure 150 foot-candles at 10 feet:
Very straightforward.
The Question
While gathering data for preparation for the article on
the Shadow Factor (TV Technology, 11/07/2012), it seemed an ideal
time to determine the answer to a question that I had always wondered about.
If the Inverse Square Law should not be applied at distances closer than,
say, ten times the maximum size of the source, what is the relationship
between the measurement distance and the incident light at points close to
source, within the ten times limit? Could it perhaps magically approach a
linear relationship? I must warn the reader at this point that the results I
obtained were, disappointedly, not as neat and orderly as a “Law” should
dictate.
The Method
To attempt to answer my question, I used a conventional 4KW Super Softlite for the test. The aperture of the fixture is 30 inches by 36 inches, a relatively large source. I measured the foot-candles at the center of the beam of the instrument every two feet from the front of the fixture from 2 to 49 feet. Then I plotted the data on a log-log scale graph. First, I choose to plot distances greater than ten times the instrument’s width in order to be as conservative as possible, in other words, distances greater than 30 feet (ten times the three foot width of the Softlite). The resultant curve was, happily, a straight line when plotted in the log/log scale indicating that a simple power function of the form:
could describe the curve (k being a constant). If the Square Law is, in truth, a law, the value of the exponent, should be -2 in this case as the measuring distance is greater than 10 times the fixture’s aperture. Disappointingly, when I applied the curve fitting ability of my spread sheet application, I came out with an exponent of -1.828, close, but not -2.000. This is shown in Graph (1). I have included an idealized plot (in red) that shows what the curve would look like if the Square Law represented the data.
Checks and Balances
At this point, I thought that I’d better check to see if I could find any published photometric data for similar large sources to compare. What I found was interesting. One reputable company just gave a figure for candlepower and two values for illuminance at selected distances, the illuminance obviously calculated from the candlepower listed using the Inverse Square Law. I think this was in order to simplify the data, as the casual user may be only interested in the order of magnitude of the illuminance rather than a value carried out to ten places.
The manufacturer’s data for the Softlite that I used did not have as many measurement distances as were in my test, but I was relieved to see that that data did not obey the square law as well. The exponent was -1.919 compared to -1.954 that I calculated for similar range of distances with my data. I could only find one other set of data for a large source. These readings were extensive except that they were only in range of less than ten times the width, in other words where the inverse square law should not be applied and revealed an exponent of -1.869 which I felt gave some credibility to the data I acquired.
The Answer
Going back to the original problem of what function could define the illuminance where we are in the unacceptable distance range. Referring to graph (2), I have split my measurements into three sections in order to see if a trend exists. Success at last. You will notice that the exponent of the function decreases from the nominal theoretical value of -2.0 as we measure closer to the lamp. In my case the three ranges gave exponents of decreasing values as the distance decreases from -1.954 to -1.895 and finally, -1.583. [I would guess that the exponent might actually be -1.0 at some point very close, perhaps a millimeter or so, from the fixture.]
As a result of my efforts, I think I can safely make the following statements with some degree of conviction. (1) The Inverse Square Law may be used be to approximate illuminance at other distances where illuminance is known at one distance, if all distances are greater than ten times the source’s maximum dimension. (2) Where the distances are less than ten times the large source’s maximum dimension, the Inverse Square Law does not apply precisely (but can be used to yield a useful guess). Not too scientific but no one said lighting is a science.
Bill Klages would like to extend an invitation to all the lighting people out there to give him your thoughts at billklages@roadrunner.com