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For the photoelectric emission from cesium, show that wave theory prodicts that (i) maximum kinetic energy of the photoelectrons (K_(max)) depends on the intensity I of the incident light (ii) K_(max) does not depend on the frequency of the incident light and (iii) the time interval between the incidence of light and the ejection of photoelectrons is very long. (Given : The work function for cesium is 1.90 eV and the power absorbed per unit area is 1.60 xx 10^(-6) Wm^(-2) which produces a measurable photocurrent in cesium.) |
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Answer» Solution :According to wave theory, the energy in a light wave is spread out uniformly and continuously over the wavefront. For the sake of simplicity, the following assumptions are made. (a) Light is absorbed in the top atomic layer of the metal (b) For a given element, each atom absorbs an equal amount of energy and this energy is proportional to its cross-sectional area A (c) Each atom gives this energy to one of the electrons. The energy absorbed by each electron in time t is given by E = IAt With this energy absorbed, the most energetic electron is released with `K_(max)` by overcoming the surface energy barrier or work function `phi_(0)` and this is expressed as `K_(max) = IAt - phi_(0)` Thus, wave theory predicts that for a unit time, at low light intensities when `IA lt varphi_(0)`, no electrons are emitted. At higher intensities, when `IA ge phi_(0)`, electrons are emitted. This implies that higher the light intensity, greater will be `K_(max)`. Therefore, the predictions of wave theory contradict experimental observationa at both very low and very high light intensities. `K_(max)` is dependent only on the intensity under given conditions - that is, by suitably increasing the intensity, one can produce photoelectric effect even if the frequency is less than the threshold frequency. So the concept of threshold frequency does not even exist in wave theroy. According to wave theory, the intensity of a light wave is proportional to the square of the amplitude of the electric field `(E_(0)^(2))`. The amplitude of this electric field increases with incresing intensity and imparts an increasing acceleration and kinetic energy to an electron. Now I REPLACED with a quantity proportional to `E_(0)^(2)` in equation (1). This means that `K_(max)` should not depend at all on the frequency of the classical light wave which again contradicts the experimental results. If an electron accumulates light energy just enough to overcome the work function, then it is ejected out of the atom with zero kinetic energy. Therefore, from equation (1), `0 = IAt - phi_(0)` `t = (phi_(0))/(IA) = (phi_(0))/(I(PI r^(2)))` By taking the atomic radius `r = 1.0 xx 10^(-10)` m and substituting the given values of I and `phi_(0)`, we can estimate the time interval as `t = (1.90 xx 1.6 xx 10^(-19))/(1.60 xx 10^(-6) xx 3.14 xx (1 xx 10^(-10))^(2)) = 0.61 xx 10^(7) s ~~ 71` days Thus, wave theory predicts that there is a large time GAP between the INCIDENCE of light and the ejection of photoelectrons but the experiments show that photo emission is an instantaneous process. |
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