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**Intensity, Distance and the Inverse Square Law**

My wife and I recently took the kids to the Exploratorium in San Francisco. This is a totally hands-on science museum covering a range of subjects: magnetism, sound, gravity, structural frequencies, and light. One of the exhibits covered the inverse square law of light. That got me thinking how customers often ask the Opto Squad to provide optical data transmission over long distances using a single emitter and detector. If the light is modulated, for example at 38 kHz, we can easily reach 30 meters using a TSOP3238 IR receiver and a TSAL6400 emitter while transmitting small bits of data. But if not modulated, we need to explain why the range will be relatively short; we explain the inverse square law of light transmission. I’m going to try to replicate the Exploratorium exhibit to demonstrate this.

**Set Up**

The light source is enclosed in a box with one side perforated with 256 holes in a 16 x 16 square. Light from each of these holes is a point source; it spreads out like an expanding sphere. I like to visualize a balloon blowing up. The light hits the sliding display surface that has a box drawn on the side facing the light source.

**Distance 1**

When the slide is positioned at Distance 1, all 256 points of light fit nicely inside the square box. It doesn’t matter what the actual distance from the light source to the display surface is because we are only interested in what happens as the surface is moved away from the light source.

**Distance 2**

The display surface is now twice as far from the light source. We know from experience that the farther we get from a light source, for example a street lamp, the less intense the light becomes. If we look at the number of light points in the box, we begin to see why. Much less light hits in the square, in fact exactly ¼ the amount of light than at Distance 1; 64 points of light. We doubled the distance and got ¼ the intensity. 2 squared equals 4, take the inverse and multiply it by the 256 points of light at Distance 1.

**Distance 4**

The display surface is now four times as far from the light source compared to Distance 1. What is the intensity of light in the box? Again, much less light hits in the square, in fact 1/16^{th} the amount of light than at Distance 1; 16 points of light. We quadrupled the distance and got 1/16^{th }the light. 4 squared equals 16, take the inverse and multiply it by the 256 points of light at Distance 1.

**I=P/A=P/4πR ^{2}**

The intensity of light drops off at 1/d^{2}, where d is the distance from the light source. The effects of gravity are also based on the inverse square law. The pull of the earth’s gravity drops off at 1/r^{2}, where r is the distance from the center of the earth. Sound, radiation, and electric force too.

It is something to keep in mind when you want to communicate over long distances using light. But don’t let it stop you. Just know that we have lots of tools to help you get the job done. We’re not called the Opto Squad for nothing.

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