Problem — Ranking intensities at different distances

Problem

An isotropic point source of power $P$ radiates in vacuum. Rank from most to least intense the points $A$ ($r=2$ m), $B$ ($r=4$ m), $C$ ($r=8$ m), and give the relative ratios.

Solution

Intensity law. For an isotropic source, the power is spread over spheres of area $4\pi r^2$:

$I=\frac{P}{4\pi r^2}\propto\frac{1}{r^2}$

Ratio of distances. $r_A:r_B:r_C=2:4:8=1:2:4$; hence the intensities go as $1/r^2$:

$I_A:I_B:I_C=\frac{1}{1}:\frac{1}{4}:\frac{1}{16}$

Doubling the distance divides the intensity by $4$: $B$ is a quarter of $A$, $C$ a sixteenth of $A$.

$\ev{A>B>C}$

Collegamenti

Argomenti: Onde elettromagnetiche Concetti: Intensità di un’onda Competenze: Ragionamento di ranking