11.5 In an experiment on photoelectric effect, the slope of the cut-off voltage versus frequency of incident light is found to be 4.12 × 10–15 V s. Calculate the value of Planck’s constant.

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.

11.2 The work function of caesium metal is 2.14 eV. When light of frequency 6 ×1014Hz is incident on the metal surface, photoemission of electrons occurs. What is the (a) maximum kinetic energy of the emitted electrons, (b) Stopping potential, and (c) maximum speed of the emitted photoelectrons?