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On what Factors the Value of Stopping Potential Depends? What are the Characteristics of the Photoelectric Effect?
Observation of Hertz and Contemporary Scientist: In 1887 German scientist Hertz observed that when ultraviolet rays fell on the negative electrode of a discharge tube, electric discharge occurred easily. Subsequently in Hallwachs experiment two zinc plates were placed in an evacuated quartz bulb and when ultraviolet rays fell on the plate connected to the negative terminal of the battery, immediately a current was found to flow in the circuit. But when the ultraviolet rays fell on the positive plate, there was no flow of current. He also noticed that, as soon as the ultraviolet rays were stopped, the current also stopped. Hallwachs however could not explain this phenomenon.
In 1900 Lenard proved that when ultraviolet rays fell on a metallic plate, electrons were emitted from the plate and current was constituted due to the flow of electrons. Since in this case, flow of current is due to light, it is called photoelectric effect (photo = light). For this phenomenon, light of short wavelength or high frequency is more effective than light of long wavelength or low frequency. Alkali metals e.g., lithium, sodium, potassium etc., exhibit photoelectric effect even in ordinary visible light.
Lenord’s experiment: G is an evacuated glass bulb with a quartz window Q on its lower face [Fig.]. C is a metal plate, kept at potential -V. Another plate A, having a hole at its centre is kept at zero potential by earthing. Hence the potential difference between anode and cathode = 0 – (-V) = V.
Now, the cathode is illuminated by a monochromatic beam of light, entering through the window. Here ultraviolet rays or visible light of small wavelength are used according to the nature of cathode plate metal.
Suppose, due to incidence of light, cathode C emits a beam of negatively charged particles having charge -q. These charged particles are attracted towards the positive plate A. So, the kinetic energy of each charged particle, just before reaching the plate is,
\(\frac{1}{2}\)mv2 = qV or, \(\frac{q}{m}\) = \(\frac{v^2}{2 V}\) ….. (1)
where, m = mass of each charged particle and v = velocity of the charged particle.
A beam of these particles, passing through the hole of the anode are incident on the plate P, which is connected to an electrometer to detect the current. Now, between A and P, a magnetic field B is applied perpendicularly upward with respect to the plane of the paper. Due to this field the charged particles are forced to move in circular paths. By controlling the magnetic field B, the particles are made incident on the plate D where they follow a circular path of radius of curvature R. The electrometer, connected with the plate D shows the current. Here, the magnetic force, acting on each particle,
\(\vec{F}\) = -q\(\vec{v}\) × \(\vec{B}\)
As, \(\vec{v}\) and \(\vec{B}\) both are perpendicular to each other, so the magnitude of this force,
F = |\(\vec{F}\)| = qvBsin90° = qvB
This force acts as a centripetal force for the revolving particle.
∴ qvB = \(\frac{m v^2}{R}\) or \(\frac{q}{m}\) = \(\frac{v}{B R}\)
or (\(\frac{q}{m}\))2 = \(\frac{v^2}{B^2 R^2}\) …. (2)
Dividing the equation (2) by equation (1), we get,
\(\frac{q}{m}\) = \(\frac{2 V}{B^2 R^2}\) ….. (3)
The value of \(\frac{q}{m}\) can be evaluated by putting the values of V, R and B in equation (3). From the experimental result thus obtained, Lenard had shown that the value of \(\frac{q}{m}\) is the same as the specific charge of electron, \(\frac{e}{m}\) (= 1.76 × 1011C ᐧ kg-1) as previously known. From this result, it could be concluded that the emitted negative charged particles from the cathode are electrons.
Photoelectric emission: Emission of electrons from matter (metals and non-metallic solids) as a consequence of absorption of energy from electromagnetic radiation of very short wavelengths (such as visible and ultraviolet radiation), is called photoelectric emission.
Electrons emitted in this manner, are called photoelectrons.
With proper arrangements1 the motion of photoelectrons can be made unidirectional. The stream of unidirectional photoelectrons thus produced, develops a current, namely, photoelectric current.
Work function: The minimum energy required to remove an electron from the surface of a particular substance to a point just outside the surface, is called the work function of that substance.
Here the final position of electron is far from the surface on the atomic scale but still close to the substance on macroscopic scale.
Work function depends only on the nature of the metal and is independent of the method of acquiring energy by the electron.
Work function is measured in electronvolt. Alkali metals like sodium, potassium have work functions lower than that of other metals but, nowadays, for photoelectric emission, suitable alloys are mostly used.
Demonstrative experiment: An evacuated glass bulb G with a quartz window Q, as shown in Fig. is used. Through the window, monochromatic beam of light is incident on a plate T that can emit electrons. Plate T is generally coated with an alkali metal (like sodium or potassium). When plate C is kept at a positive potential with respect to T, it attracts photo-electrons emitted from T. Hence a current is set up in the circuit which can be recorded by the gal-vanometer G’. T is called photo-cathode and C is called anode.
Rheostat Rh, in series with battery B, can be used to increase or decrease the potential difference V. Using the commutator C’, C can also be kept at negative potential with respect to T.
Ampere-Volt Characteristics : Stopping Potential
Keeping frequency of light constant: In Fig., graphs are drawn showing the dependence of I on V; where I = photoelectric current and V = potential difference between anode and cathode. Here a monochromatic light is used so that the frequency (f), of the incident light remains constant. 1 and 2 in this graph, represent I-V characteristics for different intensities of incident light. Detailed study of this graph reveal the following facts.
i) Saturation current: Characteristic curves become horizontal for higher values of V. This shows that saturation current has been achieved. So, all the electrons, emitted by the photocathode, have been collected by the anode.
ii) Effect of intensity of incident light: At constant V, with decrease in intensity, i.e., the brightness of the incident monochromatic light, the photoelectric current also decreases. Photoelectric current is directly proportional to the intensity of the incident light.
iii) Stopping potential: When a negative potential is applied to anode with respect to the photocathode, photoelectric current does not stop but decreases gradually with increase in negative potential on C. This indicates that photoelectrons possess some initial kinetic energy due to which they can reach the anode, overcoming the repulsive force of negative potential. With increase in negative potential of anode, photoelectric current becomes zero ultimately. The negative potential at this stage is called the stopping potential, or cut-off potential, or cut-off voltage, V0 [Fig.].
Definition: The minimum negative potential of anode with respect to photocathode, for which photoelectric current becomes zero, is called stopping poential.
Value of stopping potential depends on two factors:
1. Nature of surface of the photocathode and
2. frequency of the incident light [Fig.]. Analysing graphs 1 and 2 in Fig., we see that, stopping potential V0 does not depend on the intensity of incident light. Increase in intensity of incident light only increases the value of saturation current.
Keeping Intensity of light constant: In Fig. graphs are drawn showing the dependence of I on V; where I = photoelectric current and V = potential difference between anode and cathode. Lights of different frequencies but of the same intensity are used as incident light on cathode. In this figure, graphs 1 and 2 represent the I-V characteristic curves for different frequencies.
In this case, as the frequency of incident light on photocathode increases value of stopping potential also increases and vice versa. But saturation current is independent of frequency of light.
Relation between kinetic energy of photoelectrons and stopping potential: When anode potential becomes equal to the stopping potential V0, photoelectrons with even the highest kinetic energy, cannot reach the anode. Hence, the maximum kinetic energy of photoelectron (Emax) = loss of energy of electron for overcoming negative potential V0. When an electron of charge e overcomes a negative potential V0, loss of energy of electron = eV0. Thus,
Emax = eV0 …… (1)
Also, if maximum initial velocity of electron is vmax, then
Emax = \(\frac{1}{2}\)mv2max (m = mass of electron)
∴ \(\frac{1}{2}\)mv2max = eV0 or, vmax = \(\sqrt{\frac{2 e V_0}{m}}\)
Clearly, the maximum kinetic energy of electron is independent of intensity of light.
Numerical Examples
Example 1.
Stopping potential for a monochromatic light of a metal surface is 4V. What is the maximum kinetic energy of photoelectrons.
Solution:
From the relation, Emax = eV0 we can say that, when stopping potential is 4V, maximum kinetic energy, Emax = 4 eV.
Example 2.
For a metal surface, ratio of the stopping potentials for two different frequencies of incident light is 1 : 4. What is the ratio of the maximum velocities in the two cases?
Solution:
Stopping potential ∝ maximum kinetic energy. Again maximum kinetic energy ∝ (maximum velocity)2. Hence, stopping potential ∝ (maximum velocity)2. If v1 and v2 are the maximum velocities in the two given cases, respectively then,
\(\frac{1}{4}\) = \(\left(\frac{v_1}{v_2}\right)^2\), i.e., v1 : v2 = 1 : 2
Threshold Frequency or Cut-off Frequency
Frequency versus stopping potential graph: In photoelectricity, neither stopping potential nor maximum energy of photoelectron is a constant quantity. Magnitudes of both V0 and Emax depend on
- the frequency of the incident light and
- the nature of the surface of the substance used.
Relation between frequency and stopping potential for different metals is shown graphically in Fig.
Characteristics of the graph: For each substance, there is a certain frequency f0 of incident light, for which the maximum kinetic energy of photoelectrons becomes zero. In other words, there is no emission of photoelectrons. Hence, whatever may be the intensity of incident radiation, no electron can leave the metal surface for light incident with frequency equal to or less than f0 i.e., photoelectric emission stops. This f0 is the threshold frequency.
Definition: The minimum frequency of incident radiation which can eject photoelectrons from the surface of a substance, is called the threshold frequency for that substance.
The maximum wavelength corresponding to the minimum frequency f0 is called the threshold wavelength. It is given by,
λ0 = \(\frac{c}{f_0}\) [c = speed of light]
Discussions:
i) From Fig. we see that, the stopping potential or maximum kinetic energy increases as the frequency of incident radiation increases. Hence in practice, almost in all cases ultraviolet rays are used, as the frequency of ultraviolet ray is much more than that of visible violet ray.
ii) Alkali metals (e.g., sodium, potassium, caesium, etc.) emit photoelectrons even for comparatively low frequency light.
Characteristics of Photoelectric Effect
- Photoelectric current is directly proportional to the intensity of the incident light.
- Maximum velocity or kinetic energy of photoelectron is independent of the intensity of incident light. On the other hand, maximum velocity or kinetic energy increases with increase in frequency of the incident light.
- For a given material, there exists a certain minimum frequency (threshold frequency, f0) of incident light below which no photoelectrons are emitted. Photelectric effect is usually prominent in the range of frequencies of yellow to ultraviolet radiation.
- Threshold frequency is different for different materials.
- Photoelectrons emitted from surface of a substance can have any velocity between zero and maximum velocity.
- Emission of a photoelectron is an instantaneous process, which means that photoelectrons are emitted as soon as light falls on the metal surface. There is practically no time gap between these two incidents.
- Emission of photoelectrons makes the rest of the surface very slightly positively charged (this principle is followed in making photovoltaic cells).
- Emission of electrons in photoelectric effect does not depend on the temperature of the surface.