Estimate the following the following two numbers should be interesting. The first number will tell you why radio engineers do not need to worry much about a photons. The second number tells you why our eye can never "count photon" even in barely detectable light.
(i) The number of photons emitted per second by a MW transmitter of 10kW power emitting radiowaves of length 500m.
(ii) The number of photons entering the pupil of our eye per second corresponding to the minimum intensity of white light that we humans can perceive `(~10^(-10)Wm^(-2))`. Take the area of the pupil to be about `0.4cm^(2)`, and the average frequency of white light to be about `6xx10^(4)Hz. (h=6.6xx10^(-34)J)`

Explanation

(a) Here,
Power emitted by transmitter , P = 10 kW
`implies P=10xx10^3Js^(-1)`
`lamda=500m`
Number of photos emitted per second , n = ?
From relation `, c = v lamda`
Frequency of photon ,
`v=c/lamda=(3xx10^8)/(500)`
`=6xx10^(5)Hz`
Energy of each photon , E = hv
Number of photons.s emitted per second
`n=P/E=P/(hv)`
`=(10xx1000)/(6.62xx10^(-34)xx6xx10^(5))`
`=2.51xx10^(31)` photons per sec
`=3xx10^(31)` photons/sec
Thus, it can be seen that the energy of a radio photon is very small and the number of photons emitted per second in a radio beam is very larger.
So , there is almost no error involved in ignoring the existence of minimum quantum of energy (photon) and treating the total energy of a radio wave as continuous .
(b) Give, intensity , `I=10^(-10) W//m2`
Area of pupil , `A = 0.4 cm^2`
Frequency v = `6xx10^(14)Hz`
Let n be the number of photons entering the pupil per units time per unit area.
Energy of each photon , E = hv
`=6.63xx10^(-34)xx6xx10^(14)J`
`~~4xx10^(-19)J`
Total energy of n photons = `nxx4xx10^(-19)J//m^2`
Since intensity I = Energy per unit area per unit time
`implies 10^(-10) W//m^2` = Total energy of n photons
`implies10^(-10) W//m^2=nxx4xx10^(-19)J//m^2`
`=2.5xx10^(8)m^(-2)s^(-1)`
Total number of photons entering the pupil per second is
`N=nxxA`
`=2.5xx10^(8) xx0.4xx10^(-4) s^(-1) =10^(4)` photons/s
Although this number is not as larger as in (a) above it is large enough for us to never sense individual photons by our eye.