Energy Transport and the Poynting Vector — Question 2

Question 2

Spherical spreading reduces intensity faster than field amplitude

An ideal isotropic source radiates total electromagnetic power P in vacuum. A small detector is far enough away for a locally plane wave. There is no absorption.

1. Derive the distance dependence of intensity and rms electric field.

2. If rms field drops from E1 to E1/3 while the source power is unchanged, find the distance ratio and the intensity ratio.

Question 2 - Solution

Spherical spreading reduces intensity faster than field amplitude

(a) Method and calculation

$$I(r)=\frac{P}{4\pi r^2},\quad E_{\mathrm{rms}}=\sqrt{I/(\epsilon_0c)}.$$$$\boxed{E_{\mathrm{rms}}(r)=\frac1r\sqrt{\frac{P}{4\pi\epsilon_0c}}}.$$

(b) Interpretation and further calculation

$$E_2/E_1=r_1/r_2=1/3\Rightarrow\boxed{r_2/r_1=3}.$$$$\boxed{I_2/I_1=(r_1/r_2)^2=1/9}.$$

Physical check

The same total power crosses every concentric sphere; field amplitude scales as the square root of intensity.