The Inverse Square Law has to be the most exciting thing i have ever seen.... ever, infact, the excitement is almost unbearable so I suppose I better tell you about it

Basically, the inverse square law is used to work out how intense (rate of recieving energy) a wave will be depending upon how far away you are from the source and the power of the source (energy radiated per second).

The idea is that, if you have a light source in space that light gets emitted from in all directions. The light emitted at any time will move out as a sphere with a growing radius which would grow at the speed of light. Anyway, as the sphere grows larger, the surface area of the sphere is getting bigger and bigger. Therefore the light emitted from a single point source has to be spread out over a larger area. So when you are looking towards the pointsource, the further away from the source you are, the less intense the point source will seem because less light is entering your eye.

The equation to work out how intense the light will be is simply the amount of power emitted from the point source devided by the surface area of the sphere at any point.

I = P / ( 4 * π * Radius Squared)

The reason that this simple formula/idea excites me so much is simply because it affects absolutely everything we see. No other true reason.

I take no responsibility for this being proven to be completely wrong in some kind of strange extreme circumstances.

Precisely because the inverse square law is so common, is also interesting to consider the cases where it does not apply. In essence, it holds whenever something is spread evenly around a point source. It fails:

  1. When the source cannot be approximated with a point. For example, when an observer looking at an infinitely long straight glowing wire takes a step back, the light from a given part of the wire seems less intense, but at the same time more of the wire comes into view. The intensity in that case turns out to be inversely proportional to distance, i.e. I ~ r-1. The glowing filament in a light bulb is shorter than infinity, but longer than a point, so if you experimentally plot the intensity against distance you get an exponent between -2 and -1, perhaps I ~ r-1.9.

    Similarly, inside a hollow glowing ball, the light intensity from a given direction will be independent of the distance. In the case of forces, this means that there is no gravity inside a massive hollow sphere, and no electric force inside a charged shell.

  2. When the spreading is not uniform. For example, sound vibrations in a railway track are confined to the inside of the rail. They are thus kept together, and will not decrease in intensity at all. Eventually the sound will fall off for other reasons, but it is still audible over very long distances. This is why the Indians in western movies hold their ear to the track of the train they are about to ambush.

    A more esoteric example is the strong nuclear force that acts between quarks. Due to strange properties of space (lots of gluons swishing around in a less-than-zero energy state), this force is squished together like a railway track. This means that the force between a pair of quarks is independent of the distance between them, which in turn means that it is impossible to separate them completely (which would require infinite energy) - there can be no free quarks.

Some things to consider as a result of this law:

Any transmitted electromagnetic signal must be far stronger at its source/origin than is required at its destination. This is intuitive if you think about how loud one must yell to reach someone at a great distance.

What this implies is a tad scary, it means any remote control or cellphone or anything else hits you with far more energy than it does the TV or the cell-tower.

It is good to note that the energy that hits you if you were to move any distance from the transmitter would be related to the square of that distance. This is a very significant decrease. So hypothetically, if your cellphone were on the desk even two or three feet away from you, and you were using a bluetooth or wire headset to connect to the phone, you would be being bombarded by significantly less energy than would be the case if you were using the cellphone directly.

This also explains why there is serious concern that police officers could give themselves cancer by using radar guns while people who get scanned by radar guns over a large distance regularly will have (presumably) no adverse affects. (Of course in the case of police it is also an issue of prolonged exposure.) (Professor Pi says there was a study among female police officers relating breast cancer to wearing mobile phones on the chest. Among police officers there was a higher incidence of breast cancer than a control group.)

Someone please correct the logic here if i am wrong at all. Also it should be noted that the goal of this writeup is not meant to be alarmist, simply illustrative of everyday applications of the inverse square law.

thanks everyone for all the feedback and corrections

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