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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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So we have a point source, so it has no gain and radiates evenly in all directions. The result of that is that our waves will look like a sphere. Then we will have the receiver antenna into which those waves will impact.

Point transmitter emitting waves

That antenna has its own effective aperture (in the illustration, I used parabolic antenna, because the effect is most obvious with it), which is the area into which the waves from the transmitter impact. The part that impacts the Rx antenna will be received. In this problem, we don't have specifics of the Rx antenna, but we are given density of the power at the needed distance.

So how do we calculate power density? We simply take the formula for the surface of a sphere, $$P_s=4 \pi r^2 $$ Next, we spread out our 1 watt over that surface, so we have density of $$P_d=\frac{P_{Tx}}{P_s}=\frac{1\mbox{ }W}{4 \pi r^2}$$ Our problem says that $P_d=\frac{1\mbox{ } nW}{m^2}$, so we just need to solve the equation for $ r $ to get the final solution.

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