Question

Assume the local public radio station KUAF radiates 50000W of power in radio waves uniformly in all directions (in a uniform sphere). (a)What is the intensity of the waves at 1km = 1000m from the station? (b)What is the maximum amplitude of the electric field at 1km from the station? (c)What is the maximum amplitude of the magnetic field at 1km?

Answer 1

To solve this problem it is necessary to apply the equations related to intesity of plane electromagnetic wave, velocity of Speed, area of a Sphere and the permittivity of free space.

For definition we know that Intensity is equal to,

$I = \frac{P}{A}$

Where,

P = Power

A = Area (At this case sphere)

PART A) Using this definition the Intensity would be,

$I = \frac{P}{A}$

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

$I = \frac{5*10^4}{4\pi *(10^3)^2}$

$I = 3.98*10^{-3} W/m^2$

PART B) To calculate the maximum amplitud of electric field we can use the definition the intesity of plane electromagnetic wave, that is

$I = \frac{1}{2} \epsilon_0 E_0^2 c$

Where,

$\epsilon_0 =$Constant of permittivity of free space

c = Speed of light

$E_0 =$Amplitude of the electric field

Replacing we have that,

$I = \frac{1}{2} \epsilon_0 E_0^2 c$

$3.98*10^{-3}= \frac{1}{2} (8.85*10^{-12}) E_0^2 3*10^8$

Re-arrange to find $E_0,$

$E_0 = 1.731 V/m$

PART C) The amplitude of the magnetic field would be given by

$B_0 = \frac{E_0}{c}$

Where,

$E_0 =$  Amplitude of the electric field

c = Speed of light

Replacing,

$B_0 = \frac{1.731}{3*10^8}$

$B_0 = 5.77*10^{-9}T$