Problem — Wi-Fi intensity at 10 metres

Problem

A home Wi-Fi modem radiates a power $P\approx 100$ mW in an almost isotropic way (uniform in all directions). Estimate the intensity $\abs{\vv I}$ of the electromagnetic wave at a distance $r=10$ m from the modem. Is the exposure much lower than that of daytime sunlight ($\sim 10^3$ W/m²)?

Solution

Setup. With isotropic emission the power $P$ is spread uniformly over a sphere of radius $r$, whose surface is $4\pi r^2$. The intensity is power per unit surface:

$I=\frac{P}{4\pi r^2}$

Calculation. With $P=0{,}1$ W and $r=10$ m, $4\pi r^2=4\pi\cdot 100\approx 1{,}26\cdot 10^3$ m²:

$I=\frac{0{,}1}{4\pi\cdot 100}\approx 8\cdot 10^{-5}\;\text{W/m}^2$

$\ev{I\approx 8\cdot 10^{-5}\sim 10^{-4}\;\text{W/m}^2}$

Comparison. Compared with sunlight $\sim 10^3$ W/m²:

$\frac{10^3}{10^{-4}}=10^{7}$

Wi-Fi exposure is about $10^7$ times weaker than daytime sunlight: entirely negligible.

Links

Topics: Onde elettromagnetiche Concepts: Intensità di un’onda Skills: Stime di Fermi