Consider a 20W bult emitting light of wavelength 5000Å and shinning on a metal surface kept at a distance 2m. Assume that the metal surface has work function of 2eV and that each atom on the metal surface can be treated as a circular disk of radius 1.5Å.
(i) Estimate no. of photons emitted by the bulb per second. [Assume no other losses] (ii) Will there be photoelectric emission? (iii) How much time would be required by the atomic disk to receive energy equal to work function (2eV)? (iv) How many photons would atomic disk receive within time duration calculated in (iii) above? (v) Can you explain how photoelectric effect was observed instantaneously? [Hing : Time calculated in part (iii) is from classical consideration and you may further take the target of surface area say `1cm^(2)` and estimate what would happen?]

Explanation

Here, `P=20W, lambda=5000Å=5000xx10^(-10)m , d=2m, phi_(0)=2eV, r=1.5Å=1.5xx10^(-10)m`.
(i) No. of photon emitted by bulb per second is
`n'=P/((hc//lambda))=(Plambda)/(hc)=(20xx(5000xx10^(-10)))/((6.62xx10^(-34))xx(3xx10^(8)))=5xx10^(19)s^(-1)`
(ii) Energy of the incident photon `=(hc)/(lambda)=((6.62xx10^(-34))xx(3xx10^(8)))/(5000xx10^(-10)xx1.6xx10^(-19)) eV=2.48eV`
As this energy is greater than 2eV (i.e., work function of metal surface), hence photoelectric emission takes place.
(iii) Let `Deltat` be the time spent in getting the energy `phi_(0)` (=work function of metal).
`P/(4pid^(2))xxpir^(2)Deltat=phi_(0)`
or `Deltat=(4phi_(0)d^(2))/(Pr^(2))=(4xx(2xx1.6xx10^(-19))xx2^(2))/(20xx(1.5xx10^(-10))^(2))=11.4s`
(iv) No. of photons recieven by atomic disc in time `Deltat` is
`N=(n'xxpir^(2))/(4pid^(2))xxDeltat=(n'r^(2)Deltat)/(4d^(2))=((5xx10^(19))xx(1.5xx10^(-10))^(2)xx11.4)/(4xx(2)^(2))=0.80~~1`
(v) As time of emission of electrons is 11.04s, hence the photoelectric emission is not instantaneous in this problem.
In photoelectric emission, there is an elastic collision between incident photon and free electron of the metal surface, which lasts for very very short interval of time `(~~10^(-9)s)`, hence we say photoelectric emission is instantaneous.