Photoelectric Effect.
When light of sufficiently small wavelength is incident on a metal surface, electrons are ejected from the metal. This phenomenon is called the photoelectric effect. The electrons ejected from the metal are called photoelectrons. Let us try to understand photoelectric effect on the basis of the photon theory of light.
We know that there are large number of free electrons in a metal. However, these electrons are not free to leave the surface of the metal. As they try to come out of the metal. the metal attracts them back. A minimum energy, equal to the work function φ, must be given to an electron so as to bring it out of the metal.
When light is incident on a metal surface, the photons collide with the free electrons. In a particular collision, the photon may give all of its energy is more then the work function φ, the electron may come out of the metal. It is not necessary that if the energy supplied to an electron is more then φ, it will come out. The electron after receiving the energy, may lose energy to the metal in course of collision with the atoms of the metal. Only if an electron near the surface gets the extra energy and heads towards the outside, it is able to come out. If it is given an energy E which is greater than φ, and it makes the most economical use of it, it will have a kinetic energy ( E - φ ) after coming out. If it make some collision before coming out, the kinetic energy will be less than ( E - φ ). The actual kinetic energy of such an electron will depend on the total energy lost in collisions. It is also possible that the electron makes several collisions inside the metal and loses so much energy that it fails to come out. So, the kinetic energy of the photoelectron coming out may be kinetic energy of the photoelectron coming out may be anything between zero and ( E - φ ) where E is the energy supplied to the individual electrons. We can, therefore, write
Kmax = E - φ.
Work function of some photosensitive metals
Metal Work function (ev)
Cesium 1.9
Potassium 2.2
Sodium 2.3
Copper 4.5
Silver 4.7
Let monochromatic light of wavelength λ be incident on the metal surface. In the particle picture, photons of energy hc/λ fall on the surface. suppose, a particular photon collide with a free electron and supplies all its energy to the electron. the electron gets an extra energy E = hc/λ and may come out of metal. The maximum kinetic energy of this electron is, therefore,
Kmax = hc/λ-φ = hv - φ. .......(2.1)
As all the photons have the same energy hc/λ, equation (2.1) given the maximum kinetic energy of any of the ejected electrons.
Equation (2.1) is called Einstein;s photoelectric equation. Einstein, after an average academic career, put forward this theory in 1905 while working as a grade III technical officer in a patent office. He was awarded the Nobel prize in physics for 1921 for this work.
Threshold wavelength
Equation (2.1) tells that if the wavelength λ is equal to
λ༚= hc/φ,
the maximum kinetic energy is zero. An electron may just come out in this case. If λ > λ༚ , the energy hc/λ supplied to the electron is smaller than the work function φ and no electron will come out. Thus, photoelectric effect takes place only if λ ≤ λ༚ . This wavelength λ。is called the threshold wavelength for the metal. The corresponding frequency
v。 = c/λ。 = φ/h
is called the threshold frequency for the metal. Threshold wavelength and threshold frequency depend on the metal used.
Writing φ = hv。 , equation (2.1) becomes
Kmax = h(v - v。 ). ....(2.2)
Experimental Arrangement
A systematic study of photoelectric effect can be made in the laboratory with the apparatus shown in figure (2.3)
Two metal plates C and A are sealed in a vacuum chamber. Light of reasonably short wavelength passes through a transparent window in the wall of the chamber and falls on the plate C which is called the cathode or the emitter. The electrons are emitted by C and collected by the plate A called the anode or the collector. The potential difference between the cathode and the anode can be changed with the help of the batteries, rheostat and the commutator. The anode potential can be made positive or negative with respect to the cathode. The electrons collected by the anode A flow through the ammeter, batteries, etc., and are back to the cathode C and hence an electric current is established in the circuit. Such a current is called a photocurrent.
As photoelectrons are emitted from the cathode C, they move toward the anode A. At any time, the space between the cathode and the anode contains a number of electrons making up the space charge. This negative charge repels the fresh electrons coming from the cathode. However, some electrons are able to reach the anode and there is a a photocurrent. When the anode is given a positive potential with respect to the cathode, electrons are attracted toward the anode and the photocurrent increases. The current thus depend on the potential to the anode. figure (2.4) shows the variation in current with potential. If the potential of the anode is increased gradually, a situation arrives when the effect of the space charge become negligible and any electron that is emitted from the cathode is able to reach the anode.
The current then become constant and is known as the saturation current. This is shown by the part bc in figure (2.4). Further increase in the anode potential does not change the magnitude of the photocurrent.If the potential of the anode is made negative with respect to the cathode, the electrons are repelled by the anode. Some electrons go back to the cathode so that the current decreases. At a certain value of this negative potential, the current is completely stopped. The smallest magnitude of the anode potential which just stops the photocurrent, is called the stopping potential.
The stopping potential is related to the maximum kinetic energy of the ejected electrons. To stop the current, we must ensure that even the fastest electron fails to reach the anode. Suppose, the anode is kept at a negative potential of magnitude V。with respect to the cathode. As a photoelectron travels from the cathode to the anode, the potential energy increases by eV。.This is equal to the decrease in the kinetic energy of the photoelectron. as it reaches the anode, is Kmax - eV。. If the fastest electron just fails to reach the anode, we should have
eV = Kmax = hc/λ- φ
or, V。= hc/e(1/λ) - φ/e . ...(2.5)
We see that the stopping potential V。depends on the wavelength of the light and the work function of the metal. It does not depend on the intensity of light. Thus, if an anode potential of -2.0 V stops the photocurrent from a metal when a 1 W source of light is used, the same potential of -2.0 V will stop the photocurrent when a 100 W source of light of the same wavelength is udes.
The saturation current increases as the intensity of light increases. This is because, a large number of photons now fall on the meta surface and hence a larger number of electrons interact with photons. The number of electrons emitted increases and hence the current increases.
Figure (2.6a) shows plots of photocurrent versus anode potential for three difference intensities of light.
Note that the stopping potential V。is independent of the intensity of light.
The variation in stopping potential V。with 1/λ is shows in figure(2.6b) for cathodes of two different metals. From equation (2.6), the slope of each curve is
tanθ = hc/e
which is the same for all metals. Also, the curves intersect the 1/λ axis where V。is zero. Using equation (2.6), this corresponds to
hc/λ。= φ
or, 1/λ。= φ/hc
Which is inverse of the threshold wavelength.
Let us summaries the results obtained from the experiments described above.
1. When light of sufficiently small wavelength falls on a metal surface, the metal emits electrons. The emission is almost instantaneous.
2. There is a threshold wavelength λ。for a given metal such that if the wavelength of light is more than λ。, no photoelectric effect takes place.
3. The kinetic energies of the photoelectrons vary from zero to a maximum of Kmax = hc/λ - φ
with usual meaning of the symbols.
4. The photocurrent may be stopped by applying a negative potential to the anode with respect to the cathode. The minimum magnitude of the potential needs to stop the photocurrent is called the stopping potential. It is proportional to the maximum kinetic energy of the photoelectrons.
5. The stopping potential does not depend on the intensity of the incident light. This means that the kinetic energy of the photoelectrons is independent of intensity of light.
6. The stopping potential depends on the wavelength of the incident light.
7. The photocurrent increases if the intensity of the incident light is increased.
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