Chapter 24: Q38PE (page 889)

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 $r^{2}$ , the distance from the source squared. (b) Show that the magnitudes of the electric and magnetic fields are inversely proportional to r .

Short Answer

Expert verified

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

(b) Both $B_{0}$ and $E_{0}$ 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

$I = \frac{P}{A} I = \frac{P}{4 {ττr}^{2}}$

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

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

Explanation

(b)

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

$I = \frac{c ε_{0} {E_{0}}^{2}}{2} Also I = \frac{p}{4 {ττr}^{2}} Equate both the equations,$

$\frac{c ε_{0} {E_{0}}^{2}}{2} = \frac{p}{4 {ττr}^{2}} {E_{0}}^{2} = \frac{P}{2 {ττr}^{2} {cε}_{0}} E_{0} = \sqrt{\frac{p}{2 {ττr}^{2} {cε}_{0}}} E_{0} \propto \frac{1}{r} Also, from the magnetic field and electric field relationship, {cB}_{0} = E_{0} B_{0} \propto E_{0} B_{0} \propto \frac{1}{r} As a result, both B_{0} and E_{0} are proportionate to \frac{1}{r} .$