Problem — Reading an I vs r graph

Problem

An isotropic antenna radiates power $P$. A probe measures the intensities at various distances:

$r$ (m)	1	2	5	10
$I$ (W/m²)	7.96	1.99	0.318	0.0796

Verify that $I\propto 1/r^2$ by computing $I\cdot r^2$ and derive $P$.

Solution

Proportionality test. If $I\propto 1/r^2$, then the product $I\cdot r^2$ must stay constant:

$I\cdot r^2:\quad 7{,}96\cdot 1=7{,}96;\quad 1{,}99\cdot 4=7{,}96;\quad 0{,}318\cdot 25=7{,}96;\quad 0{,}0796\cdot 100=7{,}96$

The product is constant, $K\approx 7{,}96$ W: this confirms the $I\propto 1/r^2$ law typical of an isotropic source.

Finding the power. From $I=\dfrac{P}{4\pi r^2}$ it follows that $P=4\pi\,(I\,r^2)=4\pi K$:

$P=4\pi\cdot 7{,}96\approx 100\ \text{W}$

$\ev{P\approx 100\ \text{W}}$

Collegamenti

Argomenti: Onde elettromagnetiche Concetti: Intensità di un’onda Competenze: Lettura dei grafici