Estimate the following the following two numbers should be interesting. The first number will tell you why radio engieers 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 precieve `(~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

(i) Here, power of transmitter,
`P=10kW=10^(4)W`.
Total energy emitted per second =Pt.
`=10^(4)xx1=10^(4)J`
Energy of each photon.
`E=(hc)/lambda=((6.63xx10^(-34))xx(3xx10^(8)))/500`
In n is the number of photons emitted, then
`n=(10^(4))/E=(10^(4)xx500)/(6.63xx3xx10^(-26))=2.51xx10^(31)`
We note that the energy of a radio photon is exceedingly small and the number of photons emitted per second in a radio beam is enormously large. There is, therefore, negligible error involved in ignoring the existence of a minimum quantum of energy (photon) and treating the total energy of a radiowave as continuous.
(ii) Here, area of the pupil, `A=0.4cm^(2)`
`=4.0xx10^(-4)m^(2), v=6xx10^(14)Hz`,
Intensity,` I=10^(-10)Wm^(-2)` Energy of a photon,
`E=hv=(6.63xx10^(-34))xx(6xx10^(14))~=4xx10^(-19)J`
If n=number of photons falling per second per unit area. Then energy per unit area per second due to these photons=total energy of n photons
`=nxx4xx10^(-19)Jm^(-2)`
Since, intensity=energy per unit area per second
`:. 10^(-10)=nxx4xx10^(-19)`
or `n=(10^(-10))/(4xx10^(-19))=2.5xx10^(8)m^(-2)s^(-1)`
`:.` Number of photons entering the pupil per second
`=nxx area of the pupil`
`=(2.5xx10^(8))xx(0.4xx10^(-4))s^(-1)`
`=10^(4)s^(-1)`
Though this number is not large as in part (i) above, it is large enough for as never "sense" or "count" the individual photons by our eye.