Mathematics ELECTRON EMISSION AND PHOTOELECTRIC EFFECT

### Topic Covered

color{blue}{star} INTRODUCTION
color{blue}{star} ELECTRON EMISSION
color{blue}{star} PHOTOELECTRIC EFFECT
color{blue}{star} EXPERIMENTAL STUDY OF PHOTOELECTRIC EFFECT

### INTRODUCTION

color{blue} ✍️The Maxwell’s equations of electromagnetism and Hertz experiments on the generation and detection of electromagnetic waves strongly established the wave nature of light.

color{blue} ✍️Towards the same period experimental investigations on conduction of electricity (electric discharge) through gases at low pressure in a discharge tube led to many historic discoveries.

color{blue} ✍️It was found that at sufficiently low pressure of about 0.001 mm of mercury column, a discharge took place between the two electrodes on applying the electric field to the gas in the discharge tube.

color{blue} ✍️A fluorescent glow appeared on the glass opposite to cathode. The colour of glow of the glass depended on the type of glass, it being yellowish-green for soda glass. The cause of this fluorescence was attributed to the radiation which appeared to be coming from the cathode.

color{blue} ✍️By applying mutually perpendicular electric and magnetic fields across the discharge tube, J. J. Thomson was the first to determine experimentally the speed and the specific charge [charge to mass ratio (e//m)] of the cathode ray particles.

color{blue} ✍️They were found to travel with speeds ranging from about 0.1 to 0.2 times the speed of light color{purple}{(3 ×10^8 m//s)}. The presently accepted value of color{purple}{e//m} is color{purple}{1.76 × 10^(11) C//kg.}

color{blue} ✍️Further, the value of e/m was found to be independent of the nature of the material/metal used as the cathode (emitter), or the gas introduced in the discharge tube. This observation suggested the universality of the cathode ray particles.

color{blue} ✍️Around the same time, it was found that certain metals, when irradiated by ultraviolet light, emitted negatively charged particles having small speeds.Also, certain metals when heated to a high temperature were found to emit negatively charged particles. The value of e//m of these particles was found to be the same as that for cathode ray particles.

color{blue} ✍️These observations thus established that all these particles, although produced under different conditions, were identical in nature. J. J. Thomson, named these particles as electrons, and suggested that they were fundamental, universal constituents of matter.

color{blue} ✍️Later, American physicist R. A. Millikan performed the pioneering oil-drop experiment for the precise measurement of the charge on an electron. He found that the charge on an oil-droplet was always an integral multiple of an elementary charge, color{purple}{1.602 × 10^(–19) C}. Millikan’s experiment established that electric charge is quantised. From the values of charge (e) and specific charge color{purple}{(e//m),} the mass (m) of the electron could be determined.

### ELECTRON EMISSION

color{blue} ✍️We know that metals have free electrons (negatively charged particles) that are responsible for their conductivity. However, the free electrons cannot normally escape out of the metal surface.

color{blue} ✍️ If an electron attempts to come out of the metal, the metal surface acquires a positive charge and pulls the electron back to the metal. The free electron is thus held inside the metal surface by the attractive forces of the ions.

color{blue} ✍️Consequently, the electron can come out of the metal surface only if it has got sufficient energy to overcome the attractive pull. A certain minimum amount of energy is required to be given to an electron to pull it out from the surface of the metal.

color{blue} ✍️This minimum energy required by an electron to escape from the metal surface is called the work function of the metal. It is generally denoted by phi_0 and measured in eV (electron volt).

color{blue} ✍️One electron volt is the energy gained by an electron when it has been accelerated by a potential difference of 1 volt, so that 1 eV = 1.602 ×10^(–19) J. This unit of energy is commonly used in atomic and nuclear physics. The work function (phi_0) depends on the properties of the metal and the nature of its surface.

color{blue} ✍️The values of work function of some metals are given in Table 11.1. These values are approximate as they are very sensitive to surface impurities. Note from Table 11.1 that the work function of platinum is the highest color{purple}{(phi_0 = 5.65 eV)} while it is the lowest color{purple}{(phi_0 = 2.14 eV)} for caesium.

color{blue} ✍️The minimum energy required for the electron emission from the metal surface can be supplied to the free electrons by any one of the following physical processes:

color{bgreen} {(i) "Thermionic emission:"} By suitably heating, sufficient thermal energy can be imparted to the free electrons to enable them to come out of the metal.

color{green} {(ii)} "Field emission:"} By applying a very strong electric field (of the order of color{purple}{108 V m^(–1))} to a metal, electrons can be pulled out of the metal, as in a spark plug.

color{green} {(iii)} "Photo-electric emission:"} When light of suitable frequency illuminates
a metal surface, electrons are emitted from the metal surface. These photo(light)-generated electrons are called photoelectrons.

### PHOTOELECTRIC EFFECT

color{brown}bbul("Hertz’s observations")
color{blue} ✍️The phenomenon of photoelectric emission was discovered in 1887 by Heinrich Hertz, during his electromagnetic wave experiments.

color{blue} ✍️In his experimental investigation on the production of electromagnetic waves by means of a spark discharge, Hertz observed that high voltage sparks across the detector loop were enhanced when the emitter plate was illuminated by ultraviolet light from an arc lamp.

color{blue} ✍️Light shining on the metal surface somehow facilitated the escape of free, charged particles which we now know as electrons. When light falls on a metal surface, some electrons near the surface absorb enough energy from the incident radiation to overcome the attraction of the positive ions in the material of the surface.

color{blue} ✍️After gaining sufficient energy from the incident light, the electrons escape from the surface of the metal into the surrounding space.

color{brown}ulbb("Hallwachs’ and Lenard’s observations")
color{blue} ✍️Wilhelm Hallwachs and Philipp Lenard investigated the phenomenon of photoelectric emission in detail during 1886-1902.

color{blue} ✍️Lenard observed that when ultraviolet radiations were allowed to fall on the emitter plate of an evacuated glass tube enclosing two electrodes (metal plates), current flows in the circuit (Fig. 11.1).

color{blue} ✍️As soon as the ultraviolet radiations were stopped, the current flow also stopped. These observations indicate that when ultraviolet radiations fall on the emitter plate C, electrons are ejected from it which are attracted towards the positive, collector plate A by the electric field.

color{blue} ✍️The electrons flow through the evacuated glass tube, resulting in the current flow. Thus, light falling on the surface of the emitter causes current in the external circuit. Hallwachs and Lenard studied how this photo current varied with collector plate potential, and with frequency and intensity of incident light.

color{blue} ✍️Hallwachs, in 1888, undertook the study further and connected a negatively charged zinc plate to an electroscope. He observed that the zinc plate lost its charge when it was illuminated by ultraviolet light.

color{blue} ✍️Further, the uncharged zinc plate became positively charged when it was irradiated by ultraviolet light. Positive charge on a positively charged zinc plate was found to be further enhanced when it was illuminated by ultraviolet light.

color{blue} ✍️From these observations he concluded that negatively charged particles were emitted from the zinc plate under the action of ultraviolet light.

color{blue} ✍️After the discovery of the electron in 1897, it became evident that the incident light causes electrons to be emitted from the emitter plate.

color{blue} ✍️Due to negative charge, the emitted electrons are pushed towards the collector plate by the electric field. Hallwachs and Lenard also observed that when ultraviolet light fell on the emitter plate, no electrons were emitted at all when the frequency of the incident light was smaller than a certain minimum value, called the threshold frequency.
This minimum frequency depends on the nature of the material of the emitter plate.

color{blue} ✍️ It was found that certain metals like zinc, cadmium, magnesium, etc., responded only to ultraviolet light, having short wavelength, to cause electron emission from the surface.
However, some alkali metals such as lithium, sodium, potassium, caesium and rubidium were sensitive even to visible light. All these photosensitive substances emit electrons when they are illuminated by light. After the discovery of electrons, these electrons were termed as photoelectrons. The phenomenon is called "photoelectric effect."

### EXPERIMENTAL STUDY OF PHOTOELECTRIC EFFECT

color{blue} ✍️Figure 11.1 depicts a schematic view of the arrangement used for the experimental study of the photoelectric effect. It consists of an evacuated glass/quartz tube having a photosensitive plate C and another metal plate A. Monochromatic light from the source S of sufficiently short wavelength passes through the window W and falls on the photosensitive plate C (emitter).

color{blue} ✍️A transparent quartz window is sealed on to the glass tube, which permits ultraviolet radiation to pass through it and irradiate the photosensitive plate C. The electrons are emitted by the plate C and are collected by the plate A (collector), by the electric field created by the battery.

color{blue} ✍️The battery maintains the potential difference between the plates C and A, that can be varied. The polarity of the plates C and A can be reversed by a commutator. Thus, the plate A can be maintained at a desired positive or negative potential with respect to emitter C. When the collector plate A is positive with respect to the emitter plate C, the electrons are attracted to it.

color{blue} ✍️The emission of electrons causes flow of electric current in the circuit. The potential difference between the emitter and collector plates is measured by a voltmeter (V) whereas the resulting photo current flowing in the circuit is measured by a microammeter (μA).

color{blue} ✍️The photoelectric current can be increased or decreased by varying the potential of collector plate A with respect to the emitter plate C. The intensity and frequency of the incident light can be varied, as can the potential difference V between the emitter C and the collector A.

color{blue} ✍️We can use the experimental arrangement of Fig. 11.1 to study the variation of photocurrent with (a) intensity of radiation, (b) frequency of incident radiation, (c) the potential difference between the plates A and C, and (d) the nature of the material of plate C.

color{blue} ✍️Light of different frequencies can be used by putting appropriate coloured filter or coloured glass in the path of light falling on the emitter C. The intensity of light is varied by changing the distance of the light source from the emitter.

color{brown}bbul("Effect of intensity of light on photocurrent")
color{blue} ✍️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 accelerating potential fixed, the intensity of light is varied and the resulting photoelectric current is measured each time.

color{blue} ✍️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 second. This implies that the number of photoelectrons emitted per second is directly proportional to the intensity of incident radiation.

color{brown} {bbul("Effect of potential on photoelectric current")}
color{blue} ✍️We first keep the plate A at some positive accelerating potential with respect to the plate C and illuminate the plate C with light of fixed frequency n and fixed intensity I_1. 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 accelerating (positive) potential.

color{blue} ✍️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.

color{blue} ✍️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.

color{blue} ✍️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 most energetic electrons are able to reach the collector A.

color{blue} ✍️The photocurrent is found to decrease rapidly until it drops to zero at a certain sharply defined, critical value of the negative potential V_0 on the plate A. For a particular frequency of incident radiation, the minimum negative (retarding) potential V_0 given to the plate A for which the photocurrent stops or becomes zero is called the cut-off or "stopping potential."

color{blue} ✍️The interpretation of the observation in terms of photoelectrons is straightforward. All the photoelectrons emitted from the metal do not have the same energy. Photoelectric current is zero when the stopping potential is sufficient to repel even the most energetic photoelectrons, with the maximum kinetic energy color{blue}((K_(max)), so that K_(max) = e V_0)

color{blue} ✍️We can now repeat this experiment with incident radiation of the same frequency but of higher intensity color{blue}(I_2 \ \ "and" \ \ I_3 (I_3 > I_2 > I_1)).

color{blue} ✍️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 I_1, as shown graphically in Fig. 11.3.

color{blue} ✍️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.

color{blue} ✍️We now study the relation between the frequency n of the incident radiation and the stopping potential V_0. We suitably adjust the same intensity of light radiation at various frequencies and study the variation of photocurrent with collector plate potential.

color{blue} ✍️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.

color{blue} ✍️The stopping potential is more negative for higher frequencies of incident radiation. Note from Fig. 11.4 that the stopping potentials are in the order color{red}((V_0)_3 > (V_0)_2 > (V_0)_1) if the frequencies are in the order color{green}(n_3 > n_2 > n_1).

color{blue} ✍️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"
color{blue} {(i)} the stopping potential V_0 varies linearly with the frequency of incident radiation for a given photosensitive material.
color{blue} {(ii)} there exists a certain minimum cut-off frequency V_0 for which the stopping potential is zero.

color{brown} {ulbb{"These observations have two implications :"}}
color{blue} {(i)} The maximum kinetic energy of the photoelectrons varies linearly with the frequency of incident radiation, but is independent of its intensity.

color{blue} {(ii)} For a frequency a of incident radiation, lower than the cut-off frequency V_0, no photoelectric emission is possible even if the intensity is large.

color{blue} ✍️This minimum, cut-off frequency V_0 is called the threshold frequency. It is different for different metals.

color{blue} ✍️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.

color{blue} ✍️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.

color{brown} {bbul{"We now summarise the experimental features and observations"}}
color{brown} {bbul{"described in this section."}}
color{blue} {(i)} For a given photosensitive material and frequency of incident radiation m(above the threshold frequency), the photoelectric current is directly proportional to the intensity of incident light (Fig. ).

color{blue} {(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 ).

color{blue} {(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 ).

color{blue} {(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.