What is the reason for the square attenuation?

[iVenky]

I have seen this concept in many places like electrical field, gravitational field, brightness from a distant star or in satellite communication. That is, they seem to be inversely proportional to the square of the distance from the source of that field. What's the reason behind it?

 

[0xDEADBEEF]

Conservation of energy. When something like sound radiates in all directions and you draw two balls around it - an inner one and an outer one all the sound that passes through the outer ball must have passed through the inner ball before. But the surface area of the balls increases quadratically with radius, so the intensity (the energy per area) must drop as the square of the radius. The law simply states that the energy spreads out and does not get lost.

 

[iVenky]

Oh thanks. Simple yet it didn't strike me.

 

[ModusPwnd]

The same thinking can clue you into other geometries. Consider the infinite cylinder. In this case field lines can only diverge in one direction, not in two. So with this geometry we get a 1/r dependence. Consider the infinite sheet. In this case field lines cannot diverge in any direction. Here we get no drop off with distance.

 

[Emilyjoint]

It is known as the inverse square law and is a characteristic of radiation from a point source with no absorption.
It would not be true for light from a distant star if the light could be absorbed by dust in space.
It also applies to the force of gravity as distance increases from a planet.

 

[litup]

The same thinking can clue you into other geometries. Consider the infinite cylinder. In this case field lines can only diverge in one direction, not in two. So with this geometry we get a 1/r dependence. Consider the infinite sheet. In this case field lines cannot diverge in any direction. Here we get no drop off with distance.

But a sheet is by definition a two dimensional plane. If you have a field starting somewhere in it the field does drop of with distance because a circle of radius 1 has X amount of energy in it but a circle of radius 2X still has the same amount of energy so it has to diminish by the difference of the areas. The area of radius 1 has 3.14 (Pi) units but the area of radius 2 has 12.56 area, 4 times so it goes up by radius squared and so the energy density goes down by the same amount, conservation of energy holds up.

 

[sophiecentaur]

If the Maths is not to your taste, then the image in this link says it all, I think. (About half way down)

 

[MikeGomez]

But a sheet is by definition a two dimensional plane. If you have a field starting somewhere in it the field does drop of with distance because a circle of radius 1 has X amount of energy in it but a circle of radius 2X still has the same amount of energy so it has to diminish by the difference of the areas. The area of radius 1 has 3.14 (Pi) units but the area of radius 2 has 12.56 area, 4 times so it goes up by radius squared and so the energy density goes down by the same amount, conservation of energy holds up.

ModusPnwd is correct. There is no drop off with distance in the case of the infinite sheet.

Think about it. You can not diminish by the difference in area, because there is no difference in area (always infinite).