Question

A $1kW$ radio transmitter operates at a frequency of $880Hz$ . How many photons per second does it emit?

Answer 1

The total energy of a photon is directly proportional to its frequency. Mathematically, total energy is equal to the product of frequency and Planck's constant. In order to get the total number of electrons, we just need to divide the total energy with the energy of one photon.

Formulas used: $E = \dfrac{h}{nu}$
Where $E$ is the energy of a photon, $h$ is the Planck’s constant and $\nu$ is the frequency at which the photons are emitted.

Complete step by step answer:
We know that $1kW$ denotes the power of the transmitter, where power is energy per unit time. So, this transmitter emits $1kW = 1000W = 1000Joules/s$ of power.
First, let us find the energy possessed by a single electron. This multiplied by the number of photons emitted per second will give us the total energy per second, which is equal to $1kW$ . For finding energy of a photon, we use the formula:
$E = h\nu$
Where $E$ is the energy of a photon, $h$ is the Planck’s constant and $\nu$ is the frequency at which the photons are emitted.
We know, $h = 6.625 \times {10^{ - 34}}J/s$ and the frequency is given as $880Hz = 880{s^{ - 1}}$
$\Rightarrow E = 6.625 \times {10^{ - 34}}J{s^{ - 1}} \times 880{s^{ - 1}}$
On solving this, we get:
$E = 5830 \times {10^{ - 34}} = 5.83 \times {10^{ - 31}}.Joules$
This is the energy possessed by a single photon. This multiplied by the number of photons emitted per second will give us the total energy per second.
Therefore, to get the total number of photons, we should divide the total energy with the energy of a single photon.
Number of photons emitted per second $= \dfrac{{1000J{s^{ - 1}}}}{{5.83 \times {{10}^{ - 31}}J}}$
On solving this, we get:

Number of photons emitted per second $= 171.526 \times {10^{31}} = 1.71526 \times {10^{33}}$ photons per second

Note: Note that Hertz ( $Hz$ ) is the SI unit of frequency, and is defined as the number of cycles per second, while $W$ is the SI unit of power and is defined as energy per unit time ( $J/s$ ). Radio transmitters transmit radio waves, which are a part of the electromagnetic spectrum and whose basic constituents are photons. The total energy of the wave is distributed equally among all the photons present in the wave, and is directly proportional to the frequency of transmission.