Question

Suppose a source of electromagnetic waves radiates uniformly in all directions in empty space where there are no absorption or interference effects. (a) Show that the intensity is inversely proportional to ${\mathbf{r}}^{\mathbf{2}}$, the distance from the source squared. (b) Show that the magnitudes of the electric and magnetic fields are inversely proportional to r .

Short answer

(a) The intensity is found to be inversely proportional to${\mathrm{r}}^2$ .

(b) Both${\mathrm{B}}_0$ and${\mathrm{E}}_0$ are proportionate to $\frac{1}{r}$.

Step-by-step solution

Define electromagnetic waves

Electromagnetic waves, or EM waves, are waves that are formed when an electric field and a magnetic field vibrate together.

Explanation

(a)

The power transferred per unit area is defined as the wave's intensity, which is provided by

$$\begin{gathered} \mathrm{I}=\frac{\mathrm{P}}{\mathrm{A}} \\ \mathrm{I}=\frac{\mathrm{P}}{4{\mathrm{\tau \tau r}}^2} \end{gathered}$$

Or

role="math" localid="1655961116164" $I\propto \frac{1}{r^2}$

As a result, the intensity is found to be inversely proportional to ${\mathrm{r}}^2$.

Explanation

(b)

The formula also gives intensity in terms of electric field and light speed

$$\begin{gathered} I=\frac{c{\epsilon }_0{E_0}^2}{2} \\ \mathrm{Also}\mathrm{I}=\frac{\mathrm{p}}{4{\mathrm{\tau \tau r}}^2} \\ \mathrm{Equate}\mathrm{both}\mathrm{the}\mathrm{equations}, \end{gathered}$$

$$\begin{gathered} \frac{c{\epsilon }_0{{\mathrm{E}}_0}^2}{2}=\frac{\mathrm{p}}{4{\mathrm{\tau \tau r}}^2} \\ {{\mathrm{E}}_0}^2=\frac{\mathrm{P}}{2{\mathrm{\tau \tau r}}^2{\mathrm{c\epsilon }}_0} \\ {\mathrm{E}}_0=\sqrt{\frac{\mathrm{p}}{2{\mathrm{\tau \tau r}}^2{\mathrm{c\epsilon }}_0}} \\ {\mathrm{E}}_0\propto \frac{1}{\mathrm{r}} \\ \mathrm{Also},\mathrm{from}\mathrm{the}\mathrm{magnetic}\mathrm{field}\mathrm{and}\mathrm{electric}\mathrm{field}\mathrm{relationship}, \\ {\mathrm{cB}}_0={\mathrm{E}}_0 \\ {\mathrm{B}}_0\propto {\mathrm{E}}_0 \\ {\mathrm{B}}_0\propto \frac{1}{\mathrm{r}} \\ \mathrm{As}\mathrm{a}\mathrm{result},\mathrm{both}{\mathrm{B}}_0\mathrm{and}{\mathrm{E}}_0\mathrm{are}\mathrm{proportionate}\mathrm{to}\frac{1}{\mathrm{r}}. \end{gathered}$$