Notes-Part-1
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Topics to be Learn : Part-1
Topics to be Learn : Part-2
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Introduction :
Quantization of energy :
- In 1900, Max Planck proposed the concept of energy quantization to explain the blackbody radiation spectrum.
- He proposed that atoms behave like tiny oscillators, emitting electromagnetic radiation in small packets of energy known as quanta rather than continuously.
- He assumed that the energy associated with a quantum of radiation (now known as a photon) is proportional to the oscillator's frequency v.
- Thus, E = nhv, where n = 1, 2, 3, 4, etc., and h is a universal constant, now called Planck's constant. For n = 1, E = hv.
- A quantum of radiation is emitted when there is a transition from higher quantized level of energy of an oscillator to lower quantized level.
Planck's model served as the foundation for Einstein's theory to explain the results of photoelectric effect experiments.
The Photoelectric Effect:
The phenomenon of emission of electrons from a metal surface when electromagnetic radiation of appropriate frequency is incident on it is known as photoelectric effect.
Hertz’s observation regarding emission of electrons from a metal surface :
Heinrich Rudolph Hertz (1857-94), a German physicist, observed that electric sparks occurred more readily when one of the electrodes of his spark-gap transmitter was exposed to ultraviolet radiation during his experiments on electromagnetic waves in 1887. This discovery was known as the Hertz effect at the time, but it is now known as the photoelectric effect.
Although Hertz did not pursue his discovery, others quickly discovered that the cause of the sparking ease was due to the emission of negatively charged particles from the irradiated electrode. Following the discovery of the electron in 1897, these particles were identified as electrons.
Photosensitive surface : The surface which emits electrons when illuminated by electromagnetic radiation of appropriate frequency is called photosensitive surface.
The material that exhibits photoelectric effect is called photosensitive material.
| Know This :
Electrical energy can be obtained from light (electromagnetic radiation) in two ways (i) photo-emissive effect as described above and (ii) photo-voltaic effect, used in a solar cell. In the latter case, an electrical potential difference is generated in a semiconductor using solar energy. |
Experimental Set-up of Photoelectric Effect:
A photoelectric cell G consists of an evacuated glass tube with a quartz window W containing a photosensitive metal plate - the emitter E and another metal plate - the collector C.
- The emitter and collector are connected to a voltage source whose voltage can be changed and to an ammeter to measure the current in the circuit.
- A potential difference of V, as measured by the voltmeter, is maintained between the emitter E (the cathode) and collector C (the anode), normally C being at a positive potential with respect to the emitter. This potential difference can be varied and C can even be at negative potential with respect to E.
- When the anode potential V is positive, it accelerates the electrons (hence called accelerating potential) while when the anode potential V is negative, it retards the flow of electrons (therefore known as retarding potential).
- Monochromatic light of variable frequency from a suitable source S (such as a carbon arc) passes through a pair of polarizers P (permitting a change in the intensity of radiation) and falls on the emitter.
- A source S of monochromatic light (light corresponding to only one specific frequency) of sufficiently high frequency (short wavelength ≤ 10-7 m) is used.
- First, the collector is kept at about 10 V positive with respect to the emitter. The photoelectric current as a function of intensity and frequency of incident radiation is studied.
- Subsequently, for different intensities and frequencies, the photoelectric current is measured as a function of the collector potential which is changed from positive, through zero, to negative.
| Note : The radiation coming out of a filter is not truly monochromatic, it lies in the wavelength range between λ and λ + Δλ that depends on the source and the filter. |
Observations from Experiments on Photoelectric Effect:
(1) Effects of incident radiation frequency and intensity on photoelectric current for a given emitter material and potential difference across the photoelectric cell :
The photoelectric cell's emitter is irradiated with monochromatic light, the frequency and intensity of which can be varied and measured continuously.
Initially, the collector is made positive in relation to the emitter, so that photoelectrons ejected from the emitter move quickly to the collector. This requires only about 10 V. The photoelectric current is investigated as a function of incident radiation intensity and frequency.
Effect of frequency: Keeping the light intensity and the accelerating potential difference V constant, the frequency of the incident radiation is varied from that of far-UV to red. It is found that for every material (usually, a metal) irradiated there is a limiting frequency below which no photoelectrons are emitted irrespective of the intensity of the radiation. This frequency, v0, called the threshold frequency or cut-off frequency, is a characteristic of the material irradiated.
The graph of photoelectric current against frequency is shown in above graph (Fig.) A and B represent two different metals. The photoelectric current is not the same in the two cases, because the intensity of light is different for different frequencies.
Effect of intensity : With an emitter of a given material, the light intensity is varied by keeping the frequency v (≥ v0) of the light and the accelerating potential difference V constant. It is found that the rate of electron emission, as indicated by the photoelectric current, is proportional to the light intensity. The graph of photoelectric current against light intensity is a straight line through (0, 0), see below graph.
If we vary either the frequency of the light or the material irradiated, only the slope of the line changes. No electrons are emitted in the absence of incident radiation.
(2) Photoelectric current variation as a function of potential difference across the photoelectric cell for incident radiation of (i) a given frequency above the threshold but different intensities (ii) a given intensity but different frequencies above the threshold :
(i) The potential difference (p.d.) across the photoelectric cell is varied keeping both the frequency v(≥ threshold frequency v0) and the intensity of the light constant. Starting with the collector at about 10 V positive, we reduce this potential to zero and then run it negative.
When the p.d. across the tube is 10 V or more, all the emitted electrons are accelerated and travel across the tube, constituting the saturation current for a given light intensity; an increase in the potential of the collector does not cause an increases in current. As the collector potential is reduced from positive values through zero to negative values, the tube current reduces because of the applied retarding potential.
In this case, some electrons stop and turn back before they can reach the collector. Eventually the potential difference is large enough to stop the current completely. This is called the stopping potential or cut-off potential VO. The product of the stopping potential and electronic charge, VOe, is equal to the maximum kinetic energy that an electron can have at the time of emission.
VOe = KEmax = ½ mv2max
In above Fig. I1 I2 and I3 are intensities of the incident radiation for the same frequency v (> v0). Keeping the accelerating voltage and incident frequency fixed, if the intensity of incident radiation was increased, the value of saturation current also increased proportionately, e.g., if the intensity was doubled, the saturation current was also doubled, but V0 is independent of I.
(ii) The above experiment is repeated with different light frequencies for a given emitter material and light intensity. It is found that the stopping potential increases linearly with the frequency (see below Fig.).
Therefore, when photoejection occurs for frequencies above v0, the maximum kinetic energy of the photoelectrons increases linearly with the frequency of the radiation.
(3) The effect and significance of extremely weak radiation with a frequency greater than the threshold frequency for the emitter material in a photoelectric effect experiment :
The photoelectric cell's emitter is exposed to monochromatic light with a frequency greater than the threshold frequency for the emitter material. The collector is held at 10 V positive with respect to the emitter, allowing photoelectrons ejected from the emitter to move quickly to the collector.
- The light is dimmed dramatically (i.e., the intensity is extremely weak). The number of photoelectrons emitted per unit time in this case is very low (and special techniques are required to detect them), but they are emitted almost instantly and with the same maximum kinetic energy as bright light of the same frequency.
- According to the wave theory of light, wave trains of pulsating electromagnetic field spread out from the source. Dim light corresponds to waves of small amplitudes and small energy. If dim light spreads over a surface, conservation of energy requires that the electrons must store energy over long periods of time, which can be several hours, before gathering enough energy to become free of the metal. The fact that photoelectrons appear immediately, within about 10−9s, can be explained only by assuming that the light energy is not spread over the surface uniformly as required by the wave theory, but falls on the surface in concentrated bundles.
Characteristics of photoelectric effect :
- There is a limiting frequency of incident radiation for each metal surface below which no photoelectrons are emitted. This frequency, known as the threshold frequency, is characteristic of the irradiated metal.
- The time rate of emission of photoelectrons increases in direct proportion to the intensity of incident radiation.
- At the time of emission, photoelectrons have speeds ranging from zero to a certain maximum value, which is characteristic of a given metal for a given frequency of incident radiation. The maximum kinetic energy of photoelectrons at the time of emission is independent of intensity but increases linearly with incident radiation frequency.
- Even under extremely weak irradiation, photoelectric emission from a given metal surface is almost instantaneous for incident radiation of frequency greater than or equal to the threshold frequency.
| Know This :
Threshold frequency : The threshold frequency for a given metal surface is the characteristic minimum frequency of the incident radiation below which no photoelectrons are emitted from that metal surface (whatever may be the intensity of incident 1ight). Threshold wavelength : The threshold wavelength for a given metal surface is the characteristic maximum wavelength of the incident radiation above which no photoelectrons are emitted from that metal surface (whatever may be the intensity of incident light). Stopping potential : The stopping potential is the value of the retarding potential difference that is just enough to prevent the most energetic photoelectrons from reaching the collector, thereby reducing the photoelectric current in a photocell to zero. |
Failure of Wave Theory to Explain the Observations from Experiments on Photoelectric Effect:
- Electromagnetic waves, according to the wave theory of light, carry the energy stored in oscillating electric and magnetic fields. When an electron in a substance absorbs enough energy, it should be liberated as a photoelectron. The frequency of light is irrelevant in this case. As a result, there should be no threshold frequency for electron emission. However, it has been discovered that there is a threshold frequency that is dependent on the metal.
- Experimentally, the maximum kinetic energy of photoelectrons increases linearly with the frequency of light. This cannot be accounted by the wave theory of light.
- If a light source is weak or far away from a metal surface, electron emission will not be nearly instantaneous. Because energy is spread across the wavefront by the wave theory of light, the electron may have to wait several hours or days for absorption of enough energy from the incident light. However, for an appropriate frequency of incident light, the photoelectric effect is almost instantaneous in experiments.
- Only one observation, photoelectric current α intensity of incident light can be accounted by the wave theory of light.
Typical values of work function for some common metals :
| Metal | Work function (in eV) |
| Potassium | 2.3 |
| Sodium | 2.4 |
| Calcium | 2.9 |
| Zinc | 3.6 |
| Silver | 4.3 |
| Aluminium | 4.3 |
| Tungsten | 4.5 |
| Copper | 4.7 |
| Nickel | 5.0 |
| Gold | 5.1 |
Based on the metal's work function values in the table above, we can conclude that gold will require the highest frequency of incident radiation to generate photocurrent.
Einstein’s Postulate of Quantization of Energy and the Photoelectric Equation:
Max Planck put forward the quantum theory in 1900 to explain blackbody spectrum. He proposed in the theory that the electromagnetic radiation emitted by the body is made up of discrete concentrated bundles of energy, each equal to hv, where h is a universal constant (now known as Planck's constant) and v is the frequency of the radiation.
Einstein put forth (1905) that these energy quanta, called light quanta, later called photons, interact with matter much like a particle. When a photon collides with an electron in an atom, the electron absorbs whole of the photon energy hv in a single collision or nothing. The electron uses this energy
(1) to liberate itself from the atom,
(2) to overcome the potential energy barrier at the surface thus liberating itself from the metal, and
(3) retains the remaining part as its kinetic energy.
Different electrons need different energies in the first two processes. There are some electrons which use minimum energy in the two processes, and hence come out of the metal with maximum kinetic energy. The minimum energy required, in the form of electromagnetic radiation, to free an electron from a metal is called the photoelectric work function Φ of that metal. Thus, for the most energetic photoelectrons at the time of emission,
Maximum kinetic energy of the electron = photon energy − photoelectric work function
∴ ½ mv2max = hv – Φ, ∴ hv = Φ + ½ mv2max
The above equation is called Einstein's photo-electric equation.
Light interacts with matter as concentrated bundles of energy rather than energy spread over a Huygens type wavefront. Even under weak irradiation, an electron absorbs a photon’s energy in a single collision. But the rate of incident photons in dim light being less, the chances of such absorption diminish and consequently the photoelectric current diminishes. However, a photoelectron is emitted as soon as a photon is absorbed.
Einstein's photoelectric equation and explanation of the various features of the photoelectric effect :
Einstein's photoelectric equation :
hv = Φ + ½ mv2max …. (1)
where h = Planck's constant, v ≡ frequency of the electromagnetic radiation, hv ≡ energy of the photon incident on a metal surface, Φ ≡ photoelectric work function, i.e., the minimum energy of light quantum required to liberate an electron from the metal surface, vmax and ½ mv2max ≡ the maximum speed and maximum kinetic energy of the photoelectrons at the time of emission. Φ = hv0, where v0 is the threshold frequency for the metal.
Explanation of the characteristics of photoelectric effect :
- From the above equation we find that for photoejection, hv ≥ Φ. That is, hvmin = hv0 must be equal to Φ. Hence, photoelectric effect is observed only if hv ≥ hv0, i.e., v > v0. This shows the existence of a threshold frequency v0 for which photoelectrons are just liberated from a metal surface (with zero kinetic energy). Since different metals differ in electronic configuration, the work function hv0 and, therefore, frequency v0 are different and characteristic of different metals.
- In this particle model of light, ‘intensity of incident radiation’ stands for the number of photons incident on a metal per unit surface area per unit time. As the number of photons incident on a metal per unit surface area per unit time increases, there is a greater likelihood of a photon being absorbed by any electron. Therefore, the time rate of photoejection and hence photoelectric current increases linearly with the intensity of the incident radiation (v ≥ v0).
- From Eq. (1), ½ mv2max = hv – Φ = h(v − v0), This shows that the maximum kinetic energy increases linearly with the frequency v of the incident photon (v ≥ v0) and does not depend on the time rate at which photons are incident on a metal surface.
- As the incident energy is concentrated in the form of a photon, and not spread over a wavefront, it is expected that an electron is emitted from the metal surface as soon as a photon (v ≥ v0) is absorbed. This is in agreement with the experimental observation.
Explanation for the photoelectric effect experiment's inverse linear dependence of stopping potential on incident wavelength:
We have eV0 = \(\frac{hc}{λ}-Φ\) where V0 is the stopping potential, e is the magnitude of the charge on the electron, h is Planck's constant, c is the speed of light in free space, λ is the wavelength of the electromagnetic radiation incident on a metal surface and Φ is the work function for the metal, h, c and e are constants. Φ is constant for a particular metal.
Hence, it follows that as 1/λ increases, V0 increases.
The plot of V0 verses 1/λ is linear. This is because the energy associated with a quantum of radiation (photon) is directly proportional to the frequency of radiation and hence inversely proportional to the wavelength of radiation.
Dimensions of Planck's constant :
The energy of a photon of frequency v is E = hv, where is the Planck's, constant.
∴ h = E/v = [ML2T−2]/[T−1] = ML2T−1
Planck’s constant has a value 6.626 × 10−34
Equation that relates the threshold wavelength (λ0), the wavelength of incident radiation (λ) and the maximum speed of a photoelectron (vmax) :
\(\frac{hc}{λ}=\frac{hc}{λ_0}+\frac{1}{2}mv^2_{max}\)
Is the required equation, where h is Planck's constant, c is the speed of light in vacuum (free space) and m is the mass of the electron.
Summary of analysis of observations from experiments on photoelectric effect:
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