Question

A 75 kW radio transmitter emits 550 kHz radio waves uniformly in all directions. At what rate do photons strike a 1.5-m-tall, 3.0-mm-diameter antenna that is 15 km away?

Short answer

The rate with which the photons strike the antenna is $3.3\times {10}^{20}{s}^{-1}$.

Step-by-step solution

Step 1. Given information is :Power is uniformly distributed in sphere of radius r = 15 kmLength of Antenna, l = 1.5 mDiameter of Antenna, d = 3.0 mmPower = 75 kWFrequency = 550 kHz

We need to find out the rate with which the photons strike the antenna.

Step 2. Using the expression for fraction of total power that strikes the Antenna.

$$\begin{gathered} P\text{'}=P\frac{A}{4{\mathrm{\pi r}}^2} \\ P\text{'}=Rhf \\ AistheareawhereA=dl \\ Rhf=P\frac{dl}{4{\mathrm{\pi r}}^2} \\ R=P\frac{dl}{4{\mathrm{\pi r}}^{2}hf} \\ R=\left(75\times {10}^{3}W\right)\frac{\left(3\times {10}^{-3}m\right)\left(1.5m\right)}{4\mathrm{\pi }{\left(15\times {10}^3\mathrm{m}\right)}^2\left(6.63\times {10}^{-34}\right)\left(550\times {10}^3\mathrm{Hz}\right)} \\ R=3.3\times {10}^{20}{s}^{-1} \end{gathered}$$