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How to solve this problem:

A radio receiver needs $$ 1 \textrm{ nW/}\textrm{m}^2 $$ of power density function, how far away from a 1-watt point source will it continue to work?

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The assumption is that the transmitter transmits the power in a perfect sphere. The power density is then the transmitter power divided by the surface area of the sphere at a given distance. From here you can work backwards. First find the surface area that would produce the power density. Then find the radius of a sphere with the surface area solved for previously.

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As vini_i mentioned:

Denote Power Density as S, Area as A, radius of sphere as r, and Power as P.

Then $$\frac{P}{A} = S = 1*10^{-9} = \frac{1}{4 \pi r^2}$$ Then $$r = \frac{500}{\sqrt{\pi}} ~= 159m$$

So at a maximum distance of 159 meters your radio receiver can still operate. Note that this is an upper bound obiously. So for any distance $$ r \leq 159 m$$ your radio receiver will work.

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