# Radio Antenna (Waves)

A radio antenna broadcasts a 1.0 MHz radio wave with 30 kW of power. Assume that the radiation is emitted uniformly in all directions What is the wave intensity 27 km from the antenna? What is the electric field amplitude at this distance?

The intensity of the radiation is defined as the power of the radiation over the total surface that radiation reaches at a moment in time. Since the radiation is emitted uniformly the wave front is a sphere. The area of the sphere is

$S =4*\pi*R^2 =4*\pi*27^2*10^6 =9160.88*10^6 m^2$

The intensity of the radiation is simply

$I = P/S =30*10^3/9160.88*10^6 =3.27*10^{-6} W/m^2$

The total energy (L) of the electric field (E) existent in a volume of space (V) is

$L =1/2*\epsilon*E^2*V$

where epsilon is the electric permittivity of free space $\epsilon = 8.84*10^{-12} F/m$

Also the total energy the antenna broadcasts is

$L = P*T = P/F$ where $T$ is the time period of the wave and $F$ the frequency of the wave

$P/F = 1/2*epsilon*E^2*V$

$E^2 =2*P/(F*\epsilon*V)$

$V = 4*\pi*R^3/3 =4*\pi*27^3*10^9/3 =82448*10^9 m^3$

$E^2 =2*30*10^3/(10^6*8.84*10^{-12}*82448*10^9) =8.23*10^{-5} V^2/m^2$

$E = 9.07*10^{-3} V/m$