# Photoelectric Effect (360 nm, 1.80 eV)

#### The photoelectric effect of a laser

A 360 nm laser with a power of 20 mW produces a photoelectric effect on a certain material. The work function of the anode is 1.80 eV.

a) What is the stopping potential for the electrons emitted

b) What is the current if?

i) the efficiency is 100%  (one photon produce one electron) and

ii) all electrons arrive at the cathode

c) Are these assumptions good or unrealistic?

d) What is the associated wavelength of the electrons?

You can find here an interesting lesson about the photoelectric effect.

#### The explanations

In the external photoelectric effect the photon looses its energy on the work function of the metal. The remaining portion of the photon energy equals the kinetic energy of the electron expelled from the material. If the energy of the photon is less than the work function no electron is expelled and the photoelectric effect does not take place. In this case, instead of ionizing the atom, an internal transition between two levels happens. Therefore,

Photon energy = Work function + Electron kinetic energy

$hF = W + Ek$

$hF = W +eU$

where U is the stopping potential that need to be applied to completely stop all electrons

$U =[(h*c/\lambda)-W)/ e = 6.62*10^{-34}*3*10^8/360*10^{-9}/1.6*10^{-19} – 1.8 =1.648 V$

If $N/t$ is number of photons in time unit then

$Power = Energy/time = (N/t) hF = (N/t)*(hc/\lambda)$

$N/t = P\lambda/(hc) = 0.02*360*10^{-9}/6.62*10^{-34}/3*10^8 =3.62*10^{16} photons/sec$

If 1 photon = 1 electron emitted then current is

$I = Q/t = (N/t)*e =3.62*10^{16}*1.6*10^{-19} =5.8*10^{-3} A =5.8 mA$

These are unrealistic assumption because of two main reasons:

– efficiency of photoelectric effect is usually 5-20%.

– electrons emitted in their way to cathode scatter on the molecules that remain in tube and do not reach the cathode. The vacuum in the tube is less than $10^{-9}$ Torr. (Ultra high vacuum)

De Broglie wavelength is

$\lambda = h/p = h/\sqrt{2mE} = h/\sqrt{2meU}=$

$=6.62*10^{-34}/sqrt{2*9.1*10^{-31}*1.6*10^{-19}*1.648}=9.56*10^{-10} m$