11.1 to study the variation of photocurrent with (a) intensity of radiation, (b) frequency of incident radiation, FIGURE 11.1 Experimental (c) the potential difference between the plates A and C, arrangement for study of and (d) the nature of the material of plate C. Light of photoelectric effect. different frequencies can be used by putting appropriate coloured filter or coloured glass in the path of light falling 277 Reprint 2025-26 Physics on the emitter C. The intensity of light is varied by changing the distance of the light source from the emitter. 11.4.1 Effect of intensity of light on photocurrent The collector A is maintained at a positive potential with respect to emitter C so that electrons ejected from C are attracted towards collector A. Keeping the frequency of the incident radiation and the potential fixed, the intensity of light is varied and the resulting photoelectric current is measured each time. It is found that the photocurrent increases linearly with intensity of incident light as shown graphically in Fig. 11.2. The photocurrent is directly proportional to the number of photoelectrons emitted per FIGURE 11.2 Variation of second. This implies that the number of photoelectrons Photoelectric current with emitted per second is directly proportional to the intensity intensity of light. of incident radiation. 11.4.2 Effect of potential on photoelectric current We first keep the plate A at some positive potential with respect to the plate C and illuminate the plate C with light of fixed frequency n and fixed intensity I1. We next vary the positive potential of plate A gradually and measure the resulting photocurrent each time. It is found that the photoelectric current increases with increase in positive (accelerating) potential. At some stage, for a certain positive potential of plate A, all the emitted electrons are collected by the plate A and the photoelectric current becomes maximum or saturates. If we increase the accelerating potential of plate A further, the photocurrent does not increase. This maximum value of the photoelectric current is called saturation current. Saturation current corresponds to the case when all the photoelectrons emitted by the emitter plate C reach the collector plate A. We now apply a negative (retarding) potential to the plate A with respect to the plate C and make it increasingly negative gradually. When the polarity is reversed, the electrons are repelled and only the sufficiently energetic electrons are able to reach the collector A. The photocurrent is found to decrease rapidly until it drops to zero at a certain sharply defined, critical value of the negative potential V0 on the plate A. For a particular frequency of incident radiation, the minimum negative (retarding) potential V0 given to the plate A for which the photocurrent stops or becomes zero is called the cut- off or stopping potential. The interpretation of the observation FIGURE 11.3 Variation of photocurrent with in terms of photoelectrons is collector plate potential for different straightforward. All the photoelectrons 278 intensity of incident radiation. emitted from the metal do not have the Reprint 2025-26 Dual Nature of Radiation and Matter same energy. Photoelectric current is zero when the stopping potential is sufficient to repel even the most energetic photoelectrons, with the maximum kinetic energy (Kmax), so that Kmax = e V0 (11.1) We can now repeat this experiment with incident radiation of the same frequency but of higher intensity I2 and I3 (I3 > I2 > I1). We note that the saturation currents are now found to be at higher values. This shows that more electrons are being emitted per second, proportional to the intensity of incident radiation. But the stopping potential remains the same as that for the incident radiation of intensity I1, as shown graphically in Fig. 11.3. Thus, for a given frequency of the incident radiation, the stopping potential is independent of its intensity. In other words, the maximum kinetic energy of photoelectrons depends on the light source and the emitter plate material, but is independent of intensity of incident radiation. 11.4.3 Effect of frequency of incident radiation on stopping potential We now study the relation between the frequency n of the incident radiation and the stopping potential V0. We suitably adjust the same intensity of light radiation at various frequencies and study the variation of photocurrent with collector plate potential. The resulting variation is shown in Fig. 11.4. We obtain different values of stopping potential but the same value of the saturation current for incident radiation of different frequencies. The energy of the emitted electrons depends on the frequency of the incident radiations. The stopping potential is FIGURE 11.4 Variation of photoelectric current more negative for higher frequencies of incident with collector plate potential for different radiation. Note from Fig. 11.4 that the stopping frequencies of incident radiation. potentials are in the order V03 > V02 > V01 if the frequencies are in the order n3 > n2 > n1 . This implies that greater the frequency of incident light, greater is the maximum kinetic energy of the photoelectrons. Consequently, we need greater retarding potential to stop them completely. If we plot a graph between the frequency of incident radiation and the corresponding stopping potential for different metals we get a straight line, as shown in Fig. 11.5. The graph shows that (i) the stopping potential V0 varies linearly with the frequency of incident radiation for a given photosensitive material. FIGURE 11.5 Variation of stopping potential V0 (ii) there exists a certain minimum cut-off with frequency n of incident radiation for a given photosensitive material. frequency n0 for which the stopping potential 279 is zero. Reprint 2025-26 Physics These observations have two implications: (i) The maximum kinetic energy of the photoelectrons varies linearly with the frequency of incident radiation, but is independent of its intensity. (ii) For a frequency n of incident radiation, lower than the cut-off frequency n0, no photoelectric emission is possible even if the intensity is large. This minimum, cut-off frequency n0, is called the threshold frequency. It is different for different metals. Different photosensitive materials respond differently to light. Selenium is more sensitive than zinc or copper. The same photosensitive substance gives different response to light of different wavelengths. For example, ultraviolet light gives rise to photoelectric effect in copper while green or red light does not. Note that in all the above experiments, it is found that, if frequency of the incident radiation exceeds the threshold frequency, the photoelectric emission starts instantaneously without any apparent time lag, even if the incident radiation is very dim. It is now known that emission starts in a time of the order of 10– 9 s or less. We now summarise the experimental features and observations described in this section. (i) For a given photosensitive material and frequency of incident radiation (above the threshold frequency), the photoelectric current is directly proportional to the intensity of incident light (Fig. 11.2). (ii) For a given photosensitive material and frequency of incident radiation, saturation current is found to be proportional to the intensity of incident radiation whereas the stopping potential is independent of its intensity (Fig. 11.3). (iii) For a given photosensitive material, there exists a certain minimum cut-off frequency of the incident radiation, called the threshold frequency, below which no emission of photoelectrons takes place, no matter how intense the incident light is. Above the threshold frequency, the stopping potential or equivalently the maximum kinetic energy of the emitted photoelectrons increases linearly with the frequency of the incident radiation, but is independent of its intensity (Fig. 11.5). (iv) The photoelectric emission is an instantaneous process without any apparent time lag (~10– 9s or less), even when the incident radiation is made exceedingly dim. 11.5 PHOTOELECTRIC EFFECT AND WAVE THEORY OF LIGHT The wave nature of light was well established by the end of the nineteenth century. The phenomena of interference, diffraction and polarisation were explained in a natural and satisfactory way by the wave picture of light. According to this picture, light is an electromagnetic wave consisting of electric and magnetic fields with continuous distribution of energy over 280 the region of space over which the wave is extended. Let us now see if this Reprint 2025-26 Dual Nature of Radiation and Matter wave picture of light can explain the observations on photoelectric emission given in the previous section. According to the wave picture of light, the free electrons at the surface of the metal (over which the beam of radiation falls) absorb the radiant energy continuously. The greater the intensity of radiation, the greater are the amplitude of electric and magnetic fields. Consequently, the greater the intensity, the greater should be the energy absorbed by each electron. In this picture, the maximum kinetic energy of the photoelectrons on the surface is then expected to increase with increase in intensity. Also, no matter what the frequency of radiation is, a sufficiently intense beam of radiation (over sufficient time) should be able to impart enough energy to the electrons, so that they exceed the minimum energy needed to escape from the metal surface . A threshold frequency, therefore, should not exist. These expectations of the wave theory directly contradict observations (i), (ii) and (iii) given at the end of sub-section 11.4.3. Further, we should note that in the wave picture, the absorption of energy by electron takes place continuously over the entire wavefront of the radiation. Since a large number of electrons absorb energy, the energy absorbed per electron per unit time turns out to be small. Explicit calculations estimate that it can take hours or more for a single electron to pick up sufficient energy to overcome the work function and come out of the metal. This conclusion is again in striking contrast to observation (iv) that the photoelectric emission is instantaneous. In short, the wave picture is unable to explain the most basic features of photoelectric emission.

11.8 Light of frequency 7.21 × 1014 Hz is incident on a metal surface. Electrons with a maximum speed of 6.0 × 105 m/s are ejected from the surface. What is the threshold frequency for photoemission of electrons?

11.6 EINSTEIN’S PHOTOELECTRIC EQUATION: ENERGY QUANTUM OF RADIATION In 1905, Albert Einstein (1879-1955) proposed a radically new picture of electromagnetic radiation to explain photoelectric effect. In this picture, photoelectric emission does not take place by continuous absorption of energy from radiation. Radiation energy is built up of discrete units – the so called quanta of energy of radiation. Each quantum of radiant energy has energy hn, where h is Planck’s constant and n the frequency of light. In photoelectric effect, an electron absorbs a quantum of energy (hn ) of radiation. If this quantum of energy absorbed exceeds the minimum energy needed for the electron to escape from the metal surface (work function f0), the electron is emitted with maximum kinetic energy Kmax = hn – f0 (11.2) More tightly bound electrons will emerge with kinetic energies less than the maximum value. Note that the intensity of light of a given frequency is determined by the number of photons incident per second. Increasing the intensity will increase the number of emitted electrons per second. However, the maximum kinetic energy of the emitted photoelectrons is determined by the energy of each photon. Equation (11.2) is known as Einstein’s photoelectric equation. We now see how this equation accounts in a simple and elegant manner all the observations on photoelectric effect given at the end of sub-section 281 11.4.3. Reprint 2025-26 Physics · According to Eq. (11.2), Kmax depends linearly on n, and is independent of intensity of radiation, in agreement with observation. This has happened because in Einstein’s picture, photoelectric effect arises from the absorption of a single quantum of radiation by a single electron. The intensity of radiation (that is proportional to the number of energy quanta per unit area per unit time) is irrelevant to this basic process. · Since Kmax must be non-negative, Eq. (11.2 ) implies that photoelectric emission is possible only if h n > f0 or n > n0 , where φ0 Albert Einstein (1879 – n0 = (11.3) 1955) Einstein, one of the h greatest physicists of all Equation (11.3) shows that the greater the work time, was born in Ulm, function f0, the higher the minimum or threshold Germany. In 1905, he frequency n0 needed to emit photoelectrons. Thus, published three path- breaking papers. In the there exists a threshold frequency n0 (= f0/h) for the first paper, he introduced metal surface, below which no photoelectric emission the notion of light quanta is possible, no matter how intense the incident (now called photons) and radiation may be or how long it falls on the surface. used it to explain the · In this picture, intensity of radiation as noted above, features of photoelectric effect. In the second paper, is proportional to the number of energy quanta per he developed a theory of unit area per unit time. The greater the number of Brownian motion, energy quanta available, the greater is the number of confirmed experimentally a electrons absorbing the energy quanta and greater, few years later and provided therefore, is the number of electrons coming out of a convincing evidence of the atomic picture of matter. the metal (for n > n0). This explains why, for n > n0, The third paper gave birth photoelectric current is proportional to intensity. to the special theory of · In Einstein’s picture, the basic elementary process relativity. In 1916, he involved in photoelectric effect is the absorption of a published the general light quantum by an electron. This process is1955) theory of relativity. Some of – Einstein’s most significant instantaneous. Thus, whatever may be the intensity later contributions are: the i.e., the number of quanta of radiation per unit area notion of stimulated per unit time, photoelectric emission is instantaneous. emission introduced in an Low intensity does not mean delay in emission, since(1879 alternative derivation of the basic elementary process is the same. Intensity Planck’s blackbody radiation law, static model only determines how many electrons are able to of the universe which participate in the elementary process (absorption of a started modern cosmology, light quantum by a single electron) and, therefore, the quantum statistics of a gas photoelectric current.EINSTEIN of massive bosons, and a Using Eq. (11.1), the photoelectric equation, Eq. (11.2), critical analysis of the foundations of quantum can be written as mechanics. In 1921, he was e V0 = h n – f0; for ν≥ ν0 awarded the Nobel Prize in physics for his contribution h φ0 (11.4) ν −ALBERT to theoretical physics and or V0 = the photoelectric effect. e e This is an important result. It predicts that the V0 282 versus n curve is a straight line with slope = (h/e), Reprint 2025-26 Dual Nature of Radiation and Matter independent of the nature of the material. During 1906-1916, Millikan performed a series of experiments on photoelectric effect, aimed at disproving Einstein’s photoelectric equation. He measured the slope of the straight line obtained for sodium, similar to that shown in Fig. 11.5. Using the known value of e, he determined the value of Planck’s constant h. This value was close to the value of Planck’s contant (= 6.626 × 10–34J s) determined in an entirely different context. In this way, in 1916, Millikan proved the validity of Einstein’s photoelectric equation, instead of disproving it. The successful explanation of photoelectric effect using the hypothesis of light quanta and the experimental determination of values of h and φ0, in agreement with values obtained from other experiments, led to the acceptance of Einstein’s picture of photoelectric effect. Millikan verified photoelectric equation with great precision, for a number of alkali metals over a wide range of radiation frequencies.