CHAPTER 11 - DUAL NATURE OF RADIATION AND MATTER
Photoelectric Effect
Photoelectric effect refers to the phenomenon in which electrons are emitted from a material when it is exposed to light (electromagnetic radiation) of sufficient energy. Photoelectric effect provided evidence for the quantized nature of light and supported the wave-particle duality of electromagnetic radiation. The dual nature of matter and the dual nature of radiation are important concepts states that a matter can exhibit both particle like and wave like properties.
In this article, we will read in detail about photoelectric effect, its principle, photoelectric equation, threshold energy and applications.
Table of Content
What is Photoelectric Effect
When a metal is exposed to light, the photoelectric effect occurs, in which the metal emits electrons from its valence shell. The emitted electron is known as photoelectron, and this phenomenon is commonly known as photoemission.
Wilhelm Ludwig Franz Hallwachs was the first to notice the photoelectric effect, which Heinrich Rudolf Hertz later confirmed. This phenomenon, as well as the quantum nature of light, were explained by Einstein. In 1921, Einstein was awarded the Nobel Prize for Physics for his work on the Photoelectric Effect.
Threshold Energy for the Photoelectric Effect
The photons that strike the metal’s surface must have enough energy to overcome the attractive forces that bind the electrons to the nuclei in order for the photoelectric effect to occur. The threshold energy (represented by the symbol Φ) is the least amount of energy required to remove an electron from a metal. A photon’s frequency must be identical to the threshold frequency in order for it to have the same energy as the threshold energy (which is the minimum frequency of light required for the photoelectric effect to occur). The corresponding wavelength (called the threshold wavelength) is generally denoted by the sign λth, and the threshold frequency is usually denoted by the symbol νth. The following is the link between the threshold energy and the threshold frequency.
Φ = hνth = hc / λth
Relationship between the Frequency of the Incident Photon and the Kinetic Energy of the Emitted Photoelectron
Ephoton = Φ + Eelectron
hν = hνth + 1/2 mev2
where,
- Ephoton signifies the incident photon’s energy, which is equal to hν.
- Φ signifies the metal surface’s threshold energy, which is equal to hνth.
- Eelectron is the photoelectron’s kinetic energy, which is 1/2mev2 (me = mass of electron = 9.1 x 10-31 kg).
There will be no emission of photoelectrons if the photon’s energy is less than the threshold energy (since the attractive forces between the nuclei and the electrons cannot be overcome). As a result, if ν < νth , the photoelectric effect will not occur. There will be an emission of photoelectrons if the photon frequency is exactly equal to the threshold frequency (ν=νth), but their kinetic energy will be zero.
What is a Photon?
A photon is the smallest discrete amount of electromagnetic energy, also known as a quantum. It’s the fundamental unit of all light.
Photons are continually in motion and travel at a constant speed of 2.998 x 108 m/s to all observers in a vacuum. The speed of light, indicated by the letter c. Every photon has a specific quantity of energy and momentum. The photon’s energy is provided by,
E = hν
where,
- h is the Planck’s constant. The value of the Planck constant is h=6.626×10–34 J s
- v is the frequency of the light.
The momentum of a photon is given by,
p = h/λ
where,
- λ is the wavelength of light.
- h is the Planck’s constant.
Properties of Photon
Photons have the following basic properties:
- The quantity of photons crossing an area per unit time increases as light intensity increases. It has no effect on the radiation’s energy.
- Electric and magnetic fields have no effect on a photon. It has no electrical charge.
- A photon is massless.
- It’s a sturdily constructed particle.
- When radiation is emitted or absorbed, photons can be generated or destroyed.
- During a photon-electron collision, the whole energy and momentum are conserved.
- A photon is incapable of decay on its own.
- A photon’s energy can be transferred when it interacts with other particles.
- In contrast to electrons, which have a spin of 1/2, a photon has a spin of one. Its spin axis is perpendicular to the travel direction. The polarization of light is supported by this feature of photons.
Minimum Condition for Photoelectric Effect
- Threshold Frequency (γth): The threshold frequency for the metal is the lowest frequency of incident light or radiation that will generate a photoelectric effect, i.e. ejection of photoelectrons from a metal surface. It is constant for one metal, but various metals may have varying values.
If γ = frequency of incident photon and γth= threshold frequency, then,
- If γ < γth, there will be no photoelectron ejection and, as a result, no photoelectric effect.
- If γ=γth, photoelectrons are simply expelled from the metal surface, and the electron’s kinetic energy is zero.
- If γ>γth, photoelectrons, and kinetic energy will be ejected from the surface.
- Threshold Wavelength (λth): The metal surface with the largest wavelength to incident light is known as the threshold wavelength during electron emission.
λth = c/γth
For λ = wavelength of the incident photon, then
- If λ<λth, the photoelectric effect will occur, and the expelled electron will have kinetic energy.
- If λ= λth, the photoelectric effect will be the only one that occurs, and the kinetic energy of the ejected photoelectron will be zero.
- There will be no photoelectric effect if λ>λth.
- Work Function or Threshold Energy (Φ): The work function/threshold energy is the minimum amount of thermodynamic work required to remove an electron from a conductor to a location in the vacuum just outside the conductor’s surface.
Φ = hγth = hc/λth
If E = energy of an incident photon, then
- If E<Φ, there will be no photoelectric effect.
- If E =Φ, just the photoelectric effect occurs, but the kinetic energy of the expelled photoelectron is 0.
- If E > photoelectron, photoelectron will be zero.
- If E >Φ, the photoelectric effect will occur, as will the expelled electron’s possession of the kinetic energy.
Principle of Photoelectric Effect
A metal surface is irradiated with light in the photoelectric effect, and when light falls on the metal’s surface, photoemission occurs, and photoelectrons are ejected from the metal’s surface. The energy of the wave’s photon is transmitted to the metal atom’s electrons, which causes the electrons to get excited and expelled with a certain velocity.
Equation of Photoelectric Effect
The photon’s energy is equal to the sum of the metal’s threshold energy and the photoelectron’s kinetic energy.
Thus, the equation of photoelectric wave is given by,
KEmax = hv–Ï•
where,
- KEmax is the maximum kinetic energy of the photoelectron
- hv is the energy of the photon.
- φ is the work function of the metal
Work function is determined by the metal in question, and it will change if the metal is changed. The work function is sometimes defined in terms of threshold frequency, which is the frequency of light for which the emitted Photoelectron’s maximal kinetic energy is zero.
Ï• = hv0
where,
- v0 is the threshold frequency.
- h is the Planck’s constant.
The maximum kinetic energy remains constant as the light intensity increases, but the value of photocurrent increases.
Characteristics Of Photoelectric Effect
- The threshold frequency varies by material; different materials have varying threshold frequencies.
- The photoelectric current is proportional to the intensity of light.
- The photoelectrons’ kinetic energy is related to the frequency of light.
- The frequency is directly proportional to the stopping potential, and the process is immediate.
Factors affecting Photoelectric Effect
The photoelectric effect depends on :
- The intensity of incident radiation.
- A potential difference between metal plate and collector.
- Frequency of incident radiation.
Applications of Photoelectric Effect
- Solar Panels use it to generate power. Metal combinations in these panels allow power to be generated from a wide variety of wavelengths.
- Sensors for motion and position: A photoelectric material is placed in front of a UV or IR LED in this case. Light is switched off when an object is placed between the LED and the sensor, and the electronic circuit recognizes a change in potential difference.
- Lighting sensors, such as those found in smartphones, allow for automatic screen brightness adjustment in response to ambient light. This is due to the fact that the quantity currently created by the photoelectric effect is proportional to the amount of light that strikes the sensor.
- Digital cameras can detect and record light because they have photoelectric sensors that respond to different colours of light.
- X-Ray Photoelectron Spectroscopy (XPS): This approach involves irradiating a surface with x-rays and measuring the kinetic energy of the electrons that are released. Important features of a surface’s chemistry, such as elemental composition, chemical composition, the empirical formula of compounds, and chemical state, can be acquired.
- In burglar alarms, photoelectric cells are utilized.
- Photomultipliers use it to detect low light levels.
- In the early days of television, it was used in video camera tubes.
- This phenomenon is used in night vision systems.
- The photoelectric effect is also useful in the research of nuclear reactions. Because released electrons tend to carry specific energy that is distinctive of the atomic source, it is used in the chemical study of materials.
Also, Check
Solved Examples on Photoelectric Effect
Example 1: Light of wavelength 4000â„« is incident on a metal plate whose work function is 2eV. What is the maximum kinetic energy of emitted photoelectron?
Solution:
The wavelength of light is λ=4000Å and work function, φ0=2eV
From the Einstein Photoelectric equation, the maximum kinetic energy of photoelectron is given by,
Kmax=(hc/λ–φ0)
where ‘h′ is Planck’s constant and ‘c′ is the speed of light in a vacuum.
Kmax=(6.6×10–34×3×108/4000×10–10–(2×1.6×10–19))
Kmax=4.95×10–19/1.6×10–19eV–2eV=1.1eV
The maximum kinetic energy 1.1eV.
Example 2: The value of retarding potential needed to stop the photoelectrons ejected from a metal surface of work function 1.2eV with the light of energy 2eV is
Solution:
Work function of the metalφ=1.2eV and energy of the photons is hν=2eV.
The maximal kinetic energy of photoelectrons is given by the Einstein photoelectric equation:
eV=hv–φ
Where ‘V′ is retarding potential or stopping potential.
h is the Planck’s constant.
φ is the work function of the metal.
V=(2eV–12eV)/e=0.8V
Thus, the retarding potential is 0.8V
FAQs on Photoelectric Effect
What is the mass of a photon?
The photon’s rest mass is zero, which indicates that if the photon is moving, it will have some momentum, which is equivalent to mass, but at rest, the photon’s mass will be zero.
What is threshold frequency?
The threshold frequency of light is the frequency at which the photoelectron’s kinetic energy is zero and it is just enough to emit photoelectron. The work function of the metal is equal to the energy associated with threshold frequency.
What is stopping potential?
When the lighted metal is retained at the cathode, the stopping potential is the lowest potential at which no photoelectrons reach the anode.
What is work function?
The minimal amount of energy necessary to extract one electron from the valence shell of a metal. It all depends on the type of metal we’re using. Only at frequencies greater than the threshold frequency does the photoelectric effect occur; if the frequency of the light wave is less than the threshold frequency, the photoelectric effect does not occur.
Experimental Study of Photoelectric Effect
Heinrich Hertz discovered the phenomena of photoelectric emission in 1887. He noticed that when the spark gap is open, it conducts electricity more easily. The ultraviolet light from the arc lamp illuminates the emitter.
During the years 1886-1902, Hallwachs and Lenard studied photoelectric emission in depth. When ultraviolet radiation is permitted to fall on the emitter plate of an evacuated glass tube containing two electrodes, current flows through the circuit, according to Lenard. The current flow stopped as soon as the UV radiations were turned off. These findings suggest that when ultraviolet light strikes the emitter plate, electrons are expelled from it, and the electric field attracts them to the positive collecting plate.
As a result, light falling on the emitter’s surface causes current to flow through the external circuit. Hallwachs and Lenard looked at how this photocurrent changed with collector plate potential, as well as light frequency and intensity. Certain metals, such as zinc, cadmium, and magnesium, were discovered to respond only to UV light with a short wavelength that caused electron emission from the surface. However, some alkali metals, such as lithium, sodium, potassium, cesium, and rubidium, are visible light sensitive. When these photosensitive compounds are irradiated by light, they all emit electrons.
What is Photoelectric Effect?
The phenomenon of emission of electrons by certain substances (say metals), when exposed to radiations of suitable frequencies is called photoelectric effect and emitted electrons are called photoelectrons.
The photoelectric effect is a one photon-one electron phenomenon. One photon cannot eject more than one photo-electron.
Experimental study of the Photoelectric effect
Heinrich Hertz discovered the phenomena of photoelectric emission in 1887. The Hertz experimental arrangement for the study of the photoelectric effect is shown in the below image.
Photoelectric Cell
It consist of,
- Evacuated glass tube
- Photosensitive plate (C)
- Metal plate (A)
- Light source (S) [Mon
- chromatic light]
- Quartz window (W)
Monochromatic light from the source of sufficiently short wavelength enters the tube through quartz window and falls on photosensitive plate C which acts as an emitter. The electrons are emitted by the plate C and are collected by the plate A (collector), the electric field is created by the battery.
The potential difference applied between the plates can be changed by the potential divider arrangement. By means of commutator key the plate A can be maintained at desired positive or negative potential with respect to Plate C.
Important Observations are,
- When light is incident on the emitter plate the photo electrons are emitted.
- The photo electrons are attracted by positive plate A.
- The emission of electrons causes flow of electric current in the circuit.
- The potential difference between the emitter and collector plates is measured by the voltmeter (V) whereas the photoelectric current flowing in the circuit is measured by microammeter (UA).
The variation in the photoelectric effects depends upon the following factors
- Frequency.
- Intensity.
- The potential difference between plates A and C.
- The nature of the material of plate C.
Effect of frequency on photoelectric current
Collector plate A is given an appropriate positive potential. The frequency of incident light is gradually increased from its smallest value while the intensity of incident radiation remains constant. It is observed that until a specific frequency u is reached, no photoelectric current is recorded [Refer below image].
Variation of Photoelectric current with a frequency of incident radiation
This frequency is a property of emitter plate C’s material, known as the threshold frequency for that metal, and is denoted by v.
Threshold frequency
The threshold frequency is the lowest frequency of incoming radiation for which photoelectrons are merely released from photosensitive material.
The threshold frequency of emitter plate C varies depending on the material. The threshold wavelength is the wavelength that corresponds to the threshold frequency (v). If λ> λo, photoemission is not possible.
Effect of intensity of light on the photoelectric current
The collector plate A is kept at a positive potential in relation to the emitter plate C, attracting electrons emitted from C to A. The intensity of light is modified while the frequency of incident radiation and accelerating potential remains constant, and the resulting photoelectric current is monitored. Photoelectric current is observed to rise linearly with light intensity, as shown in the below image.
Variation of photoelectric current with the intensity of light
The number of photoelectrons emitted per second is exactly proportional to the photoelectric current. This means that the rate at which photoelectrons are emitted is proportional to the intensity of incident energy.
Note: More photos, not more energetic photons, are produced by a brighter light.
Effect of potential difference on photoelectric current
The incident radiation threshold frequency and intensity are both kept constant at a reasonable value. Plate A’s positive potential is steadily increased, and the resulting photoelectric current is measured each time. It has been discovered that as the positive (accelerating potential) grows, so does the photoelectric current. All of the released electrons are captured by plate A at some point for a particular positive potential, and photoelectric current peaks. The photoelectric current does not rise as the positive potential of plate A is increased further. Saturation current refers to the highest value of photoelectric current.
The commutator key is now inverted, and plate A is given a negative (retarding) potential in relation to plate C. The negative potential of Plate A is gradually increased till the photoelectric current reduces to zero. The minimum negative potential V_{0} given to the plate A for which photoelectric current stops or becomes zero is called cut-off or stopping potential.
Variation of photoelectric current with collector plate potential for different intensities of incident radiation
The energy of all photoelectrons released from a metal surface is not the same. When the stopping potential is adequate to reject even the most energetic photoelectrons with the highest kinetic energy, the photoelectric current becomes zero, and
1 / 2 mv2max = eV0.
where
vmax is the maximum velocity of the photoelectron,
m is the mass of the electron, and
e is the magnitude of the charge on the electron.
When the incident radiation frequency is maintained constant and the experiment is repeated with varying strengths of the incident beam, it is discovered that V remains constant in all circumstances see above image. The stopping potential and maximum kinetic energy of photoelectrons are thus independent of incoming radiation intensity for a certain frequency of incident radiation.
Variation of photoelectric current with collector plate potential for different frequencies of incident radiation
Adjusting the same intensity of incoming radiation at various frequencies to study the fluctuation of photoelectric current with collector plate potential is illustrated in the figure. For incoming radiation of various frequencies, different values of stopping potential but the same values of saturation current are achieved. The energy of a photoelectron released is proportional to the frequency of incoming light. When the frequency of incoming radiation is increased, the stopping potential becomes more negative.
As shown, the below graph between incoming radiation frequency and related stopping potential for various metals is made up of straight lines.
Variation of stopping potential with the frequency of incident radiations
The graph shows that, for a particular photosensitive material, the stopping potential varies linearly with the frequency of incoming radiation, and there is a minimum cut-off frequency v0 below which the stopping potential is zero.
Sample Questions
Question 1: Define threshold frequency and photoelectric work function?
Answer:
Threshold frequency: The threshold frequency is the lowest frequency of incoming radiation for which photoelectrons are merely released from photosensitive material.
Photoelectric work function: The photoelectric work function is the threshold energy required by the radiation or photons that are incident on the surface of the metal. We can use the term radiation interchangeably with photons in this case.
ϕ=hν0
where h is the Planck’s constant and ν0 is the threshold frequency.
Question 2: What is the photoelectric effect? State and explain its characteristics?
Answer:
The phenomenon of emission of electrons by certain substances (say metals), when exposed to radiations of suitable frequencies is called photoelectric effect and emitted electrons are called photoelectrons. The photoelectric effect is a one photon-one electron phenomenon. One photon cannot eject more than one photo-electron.
Characteristics:
- For a given photosensitive material, there exists a certain minimum cut-off frequency of the incident radiation, called threshold frequency v0 below which no emission of photoelectrons takes place. The threshold frequency is different for different metals.
- For a given photosensitive material and frequency of incident radiation (above threshold frequency), the photoelectric current is directly proportional to the intensity of incident light.
- Above the threshold frequency v0, the maximum kinetic energy of the emitted photoelectrons increases linearly with the frequency of the incident radiation, but is independent of intensity of incident radiation.
- The emission of photo electron is an instantaneous process. There is no time lag between the irradiation of the metal surface and P emission of photoelectrons.
Question 3: Define threshold wavelength?
Answer:
The wavelength (λ0) corresponding to the threshold frequency v0, is called the threshold wavelength.
If c is the velocity of light, then
v0 = Ï• / h
For the photoelectric emission the wavelength of incident light must be less than λ0.
Question 4: What is the mass of a photon?
Answer:
The photon’s rest mass is zero, which indicates that if the photon is moving, it will have some momentum, which is equivalent to mass, but at rest, the photon’s mass will be zero.
Question 5: Define stopping potential?
Answer:
The magnitude of the retarding potential for which the photoelectric current is zero is called the stopping potential (Vs)
The value of the stopping potential is a measure of the maximum kinetic energy for the photoelectrons.
Work done on electron by,
Vs = Max K.E. for the photoelectrons.
eVs = 1/2 mv2max
Question 6: What is the effect of kinetic energies on stopping potential in photoelectric emission?
Answer:
The value of the stopping potential is a measure of the maximum kinetic energy that can be possessed by a photoelectron.
The different photoelectrons have different kinetic energies. At the stopping potential, the work done by the electron against the stopping potential V is equal to the K.E. of this electron having maximum K.E.
eV = 1/2 mv2max
where V max is the maximum velocity.
Einstein’s Photoelectric Equation
Albert Einstein published an equation to explain this effect in 1905, the annus mirabilis (wonder year) of Physics. Light, according to Einstein, is a wave that interacts with matter as a packet of energy or a quantum of energy. The photon was the quantum of radiation, and the equation was known as Einstein’s photoelectric equation.
When a substance absorbs electromagnetic radiation, electrically charged particles are emitted from or inside it, resulting in the photoelectric effect. The phenomenon of emission of electrons by certain substance (metal), when it is exposed to radiations of suitable frequencies is called as photoelectric effect and emitted electrons are called photoelectrons.
The effect is frequently characterized as the ejection of electrons from a metal plate when exposed to light. The radiant energy may be infrared, visible, or ultraviolet light, X-rays, or gamma rays; the substance could be solid, liquid, or gas; and the particles discharged could be ions or electrons.
Characteristics of Photoelectric Effect:
- For each particular photosensitive material, there is a minimal cut-off frequency of the incoming radiation, known as the threshold frequency νo, below which no photo-electrons are produced.
- The threshold frequency varies depending on the metal.
- The photoelectric current is directly proportional to the intensity of incident light for a particular photosensitive material and the frequency of incident radiation (above threshold frequency).
- Above the threshold frequency ν0, the maximal kinetic energy of the emitted photo-electrons grows linearly with incident radiation frequency but is independent of incoming radiation intensity.
- The emission of photo-electrons occurs in an instant.
- There is no time lag between irradiation of the metal surface and photo-electronic emission.
Einstein`s Photoelectric Equation
In 1905, Einstein expanded Planck’s notion by deriving his own equation that correctly described the features of the photoelectric effect.
He presupposed two things.
- Radiation with frequency ν consists of a stream of discrete quanta or photons, with energy hν, where h is Planck’s constant. Photons travel at the speed of light through space.
- When photons and electrons in the emitter’s atoms collide when radiation of frequency ν is incident on a photosensitive surface. During such a collision, the photon’s whole energy is transmitted to the electron with no time lag.
A photon is not a material particle but quanta of energy. The incident photon’s absorbed energy hv by an electron is utilized in two ways. The electron uses some of its energy to break free from the atom. The minimum energy required to free-electron from a given surface is called photoelectric work function φ0 of the material of the surface.
The residual energy (h ν- φ0) emerges as electron kinetic energy. If the electron does not lose any of its energy in impact with the surface and exits with the greatest possible kinetic energy.
∴ 1 / 2 m v2max = hν – φ0 …..(1)
where,
- m = mass of the electron
- vmax = maximum velocity of electron
All the photoelectrons emitted from the metal surfaces do not have the same energy.
The photoelectric current becomes zero when the stopping potential is sufficient to repel even the most energetic photoelectrons, with the maximum kinetic energy, so that the stopping potential (in volts) is numerically equal to the maximum kinetic energy of photo-electron in eV.
1 / 2 m v2max = ev0 …..(2)
where,
- e = magnitude of the charge on an electron.
From (2) we have,
1 / 2 m v2max = hν – φ0
eV0 = hν – φ0
This equation is Einstein`s photoelectric equation.
With the aid of Einstein’s photoelectric equation, we can now describe all of the features of the photoelectric effect:
If the frequency of incident radiation is decreased, the kinetic energy of photoelectrons also decreases, and finally, it becomes zero for a particular frequency (say ν). ν is called the threshold frequency. Thus,
When ν=νo,
then
KEmax = 1/2 m V2max = 0
Therefore from equation (1), we get
0 = hν0 – φ0
hν0 = φ0
Therefore, Einstein`s equation can be written as,
1/2 m v2max = h(ν – ν0) …..(3)
From the above equation, we can say three points,
- ν > ν0, the photoelectrons are emitted with some velocity,
- ν < ν0, no photoelectrons are emitted, and
- ν = ν0, photoelectrons are just emitted with zero kinetic energy.
A more intense beam, according to quantum theory, includes a higher number of photons. As a result, the number of photon-electron collisions increases, and more photoelectrons are released. This explains why the photoelectric current increases with incoming photon intensity. Because the photoelectric work function (φ0) is constant for every given emitter, equation (3) demonstrates that the KEmax of photoelectrons grows with the frequency of incoming radiation but does not rely on intensity.
Photo-electrons are emitted as a result of electron-photon collisions. Such collisions occur as soon as the radiation strikes the photosensitive surface, and photoelectrons are released. There is no emission of photoelectrons at the incidence cut-off. As a result, the photoelectric effect occurs instantly.
Particle nature of light: Photon
The photoelectric effect, therefore, provided proof for the unusual fact that light acted as though it were formed of quanta or packets of energy, each of energy hν.
Is the light quantum of energy to be connected with a particle? Einstein discovered that the light quantum may also be related to momentum hv. A definite value of energy and momentum is a strong indication that the light quantum can be connected with a particle.
This particle was eventually given the name photon. The particle-like behaviour of light was established in 1924 by A.H. Compton’s (1892-1962) experiment on the scattering of X-rays from electrons. Einstein received the Nobel Prize in Physics in 1921 for his contributions to theoretical physics and the photoelectric effect. Millikan received the Nobel Prize in Physics in 1923 for his discoveries on the elementary charge of electricity and the photoelectric effect.
Summarized Photon picture of electromagnetic radiation
- When radiation interacts with matter, it behaves as if it were made up of particles known as photons.
- Each photon has energy E (= hu) and momentum p (= hν / c) where c is the speed of light.
- Whatever the intensity of radiation, all photons of light of a specific frequency ν or wavelength λ, have the identical energy E( = hν = hc / λ) and momentum p (= hν / c = h/λ). By raising the intensity of light of a certain wavelength, the number of photons per second crossing a given area increases, with each photon having the same energy. As a result, photon energy is independent of radiation intensity.
- Photons are electrically neutral and are unaffected by electric or magnetic forces.
- Total energy and total momentum are preserved in a photon-particle collision (such as a photon electron collision). However, the number of photons in a collision may not be preserved. The photon might be absorbed or a new photon could be produced.
Sample Problems
Problem 1: An electron is accelerated from rest through a potential difference of 500 volts. Find the speed of the electron. (Given e = 1.6 × 10-19C, m = 19 ×10-31 Kg)
Solution:
Given that,
Voltage = 500 V
e = 1.6 × 10-19 C
m = 19 × 10 Kg
KE of emitted electron = eV = 1/2 m v2
= 2 × (1.6 × 10-19) × (500) / (9.1 × 10-31 )
V = √1.758 × 1014
= 1.326 × 107 m/s
Problem 2: A metal whose work function is 4.2 eV is irradiated by radiation whose wavelength is 2000 A°. Find the maximum kinetic energy of the emitted electron.
Solution:
Given that,
φ0 = 4.2eV = 6.72 × 10-19 J
λ = 2000 A°
1/2 m v2max = hν – φ0
KEmax = hν – φ0
= (hc / λ) – φ0
= (6.63 ×10-34 × 3 × 10) / (2 × 10-7) – (6.72 × 10-10)
= (9.945 – 6.72) × 10-19
= 2.015 eV
Problem 3: If photo-electrons are to be emitted from potassium surface with a speed of 6 ×105 m/s, what frequency of radiation must be used? The threshold frequency for potassium is 4.22 ×1014 Hz.
Solution:
vmax = 6 × 105 m/s
ν0 = 4.22 × 1014 Hz
KEmax = 1/2 mv2max = h(ν – ν0)
ν = 1/2 × (mv2max / h) + ν0
= (1/2) × [(9.1 × 10-31 ×(6 × 105)2 / 6.63 × 10-34] +4.22 × 1014 Hz
= 2.47 × 1014 + 4.22 × 1014
= 6.69 × 1014 Hz
Problem 4: A photon of wavelength 3310A° falls on a photo-cathode and an electron of energy 3 × 10-19 J is ejected. if the wavelength of the incident photon is changed to 5000 A°, the energy of the ejected electron is 9.72 × 10-20 J. Calculate the value of Plank`s constant and threshold wavelength of the photon.
Solution:
λ1 = 3310 = 3.31 × 10-7 m
KEmax (1) = 3 × 10-19 J
λ2 = 5000 = 5 × 10-7 m
KEmax = 9.72 × 10-20 J= 0.972 × 10-19 J
KEmax (1) = hc/λ1 – φ0
KEmax (2) = hc/λ2 – φ0
KEmax (1) – KEmax (2) = hc (1/λ1 – 1/λ2)
3 × 10-19 – 0.972 × 10-19 = h × 3 × 108 [(1/3.31 × 10-7) – (1/5 × 10-7)]
2.028 × 10-19 = h × 3 × 1015 (1.69 / 16.55)
h = 2.028 × 10-19 × 16.55 / 3 × 1015 × 1.69
= 6.62 × 10-34 Js
Now,
KEmax (1) = hc/λ1 – φ0
φ0 = hc/λ1 – KEmax (1)
= (6.62 × 10-34 × 3 × 108 / 3.31 × 10-7 ) – 3 × 10-19
= 3 × 10-19 J
Also,
φ0 = hc / λ0
λ0 = hc / φ0
= 6.62 × 10-34 × 3 × 108 /3 × 10-19
= 6.62 × 10-7
= 6620 A°
Problem 5: The Photo-electric function for a metal surface is 2.4 eV. If the light of wavelength 5000 A° is incident on the surface of the metal, find the threshold frequency and incident frequency, Will there be an emission of photo-electrons or not?
Solution:
φ0 = 2.34eV = 3.84 × 10-19 J
λ = 5000A° = 5 × 10-7
ν = c / λ
= 3 × 108 / 5 × 10-7
= 6 × 1034 Hz
ν0 = φ0 / h
= 3.84 × 10-19 / 6.63 × 10-34
= 5.792 × 1014 Hz
As ν > ν0, photoelectric emission is possible.
Photon
Photon is a fundamental particle of electromagnetic radiation. It is a quantum of light and other electromagnetic waves. It is the force carrier of the electromagnetic force. Photons can display wave-like behaviors such as interference and diffraction. They can also exhibit particle-like behaviors such as the photoelectric effect. This is known as wave-particle duality.
Photons carry momentum and travel at the speed of light. Photons are produced through processes such as electron transitions in atoms, particle interactions, and particle-antiparticle annihilation. In the photoelectric effect, photons transfer their energy to electrons in a material, leading to the emission of photoelectrons.
Table of Content
What are Photons?
Photons are fundamental particle of electromagnetic radiation, traveling at the speed of light. It is a quantum or discrete energy packet of electromagnetic energy. Photons are massless particles that are the carriers for electromagnetic energy.
How are Photon Produced?
Photons can be produced through various processes depending on the source of electromagnetic radiation. Production of photon is explained using the analogy,
"Think of atoms like tiny solar systems: electrons orbiting around a nucleus. When we give these electrons a boost of energy, they jump to higher orbits. But they can't stay there forever - eventually, they come back down to their normal orbits. When they do, they release energy in the form of photons, which are like tiny packets of energy. Frequency of a photon is determined by the distance the electron falls, giving rise to distinctive characteristics for each photon."
Why does Photon have Momentum?
Despite lacking mass, a photon has momentum proportionate to its energy. Based on the photon's energy and frequency, the momentum can be calculated using the Planck-Einstein relation
E = hv
However, a photon cannot have any mass because it is always travelling at the speed of light, according to Einstein's calculations. However, it is evident that the photon still needs energy in order to cause the photoelectric effect. Consequently, it stands to reason that all photon energy is in the form of motion. Thus, a photon needs momentum in order to move and have energy.
Difference between Photon and Electron
Difference between Photon and Electron are as follows:
Criteria | Photon | Electron |
|---|---|---|
Definition | Photons are elementary particles that primarily function as energy carriers. | An electron is a subatomic particle found in all atoms. |
Charge | A photon particle does not have any charge. | Its has a negative charge 1.62 × 10-19 coulombs. |
Mass | Its rest mass is zero. | Mass of an electron is 9.19 × 10-31 kg. |
Speed | A photon moves at the speed of light. | Electrons are not able to travel faster than light. |
Properties of Photon
Properties of a photon include:
- Photons are quantized particles, meaning they can only exist as discrete energy packets. This plays a fundamental role in the photoelectric effect and the emission and absorption of light by atoms and molecules.
- Photons are massless particles.
- Photons are neutral particles.
- Energy of a photon is directly proportional to its frequency, as described by Planck's Equation
E = ℎν
where,
- â„Ž is Planck's Constant
- ν is Frequency
Photons carry momentum (p = h/λ) and can transfer it to other objects.
- Photons travel at the speed of light(c), approximately 3×108 meters per second in a vacuum. This constant speed is a fundamental aspect of Einstein's theory of relativity.
- Photons show characteristics of both particles and waves. This is known as the wave-particle duality.
- Photons exhibit polarization. Polarization is the orientation of oscillations of electric and magnetic fields. Polarization can be linear, circular, or elliptical, depending on the orientations and phases of the fields.
Wave-Particle Duality
Wave-Particle Duality of photons refers to the concept that photons can exhibit both wave-like and particle-like behaviours. It is the experimental setup and the type of measurement that decide whether the photon will behave as a wave or a particle.
Wave-Particle Duality challenges the classical notions of particles and waves. This pushed for a quantum theory of the behaviour of light and electromagnetic radiation. This concept has profound implications for quantum mechanics, optics, and particle physics.
Wave-Like Properties
- Photons exhibit wave-like properties in phenomena such as interference and diffraction. When multiple photons are in superposition, they can interfere with each other and form interference patterns.
For example, in the double-slit experiment, where photons are sent through two slits, they produce an interference pattern on the screen behind the slits, similar to the interference pattern produced by waves.
Particle-Like Properties
- Photons also display particle-like properties in phenomena such as the photoelectric effect. In this effect, photons interact with electrons in a material, causing them to be emitted from the material. The energy of the ejected electrons depends on the frequency (or energy) of the incident photons, not their intensity.
Additionally, photons can be detected as discrete packets of energy when they are absorbed or emitted by atoms or molecules.
Quantum Superposition
Like other quantum particles, photons can exist in a state of superposition, where they simultaneously show both wave-like and particle-like characteristics. This superposition is described by the wave function in quantum mechanics.
Electron Transitions in Atoms
When an electron in an atom transitions from a higher energy state to a lower energy state, it emits a photon with energy equal to the energy difference between the two states.
E2 - E1 = hν
This process is responsible for the emission of photons in various spectral lines observed in atomic spectra. For example, in a fluorescent light bulb, photons are produced when electrons in gas atoms transition from higher to lower energy levels.
Particle Interactions
Photons can be generated through particle interactions, such as when high-energy charged particles, like electrons or protons, collide with matter. These interactions can produce bremsstrahlung radiation (emission of photons due to the deceleration of charged particles) or synchrotron radiation (emission of photons by charged particles moving in curved paths).
Particle-Antiparticle Annihilation
Particle-antiparticle annihilation is a process in particle physics where a particle and its corresponding antiparticle collide and annihilate, resulting in the conversion of their mass into energy. This energy is often emitted in the form of photons.
2m0c2 = hν
Photoelectric Effect
Photoelectric effect is a phenomenon where electrons are ejected from a material when it is exposed to electromagnetic radiation. This occurs when photons, the particles of light, transfer their energy to electrons in the material. This minimum energy required by an electron to leave the surface of the metal is called the work function of the metal(denoted by Ï•0). The minimum frequency of light that can emit an electron from the metal surface is known as threshold frequency and is denoted by v0.
Photoelectric effect was studied experimentally in a vacuum tube, with photoelectrons emitting from the cathode and moving towards the anode. The minimum retarding potential V0 of the anode for which the photocurrent becomes zero is called the cut-off or stopping potential. The energy lost due to this stopping potential is the maximum kinetic energy of the electron.
Photoelectric effect was first successfully explained by Einstein that led him to winning the Nobel Prize in 1921 (not theory of relativity, mind you). The governing equation for all of photoelectric effect is the following
1/2 mνmax2 = eV0 = hν - ϕ0 = h(ν - ν0)
Applications of Photons
Numerous technical uses exist for photons, a few of which are covered here:
- An significant use of photons is in lasers. Photon beams in a laser beam travel at the same wavelength and in the same direction. By passing the energised electrons via an optical gain medium, like glass or a gas, is accomplished.
- In design, engineers utilise Planck's energy formula, E (= hv), to calculate the energy change resulting from photon absorption and to ascertain the frequency of light emitted from a specific photon emission.
- Single photon detection is used in a variety of hardware random number generators.
Conclusion: Photon
Photons, the fundamental particles of electromagnetic radiation are the basic packets of energy and travells at the speed of light. In this article we have describe the characteristics of photons, including their wave-particle duality, quantized nature, mass lessness, and role as carriers of electromagnetic force.
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Numericals on Photons
1. Monochromatic light of frequency 6 × 1014 Hz is produced by a laser. What is the energy of a photon in the light beam?
Solution:
To calculate energy of a photon in the light beam, we use Planck's Equation:
E = hν
Given,
Frequency(v) = 6 × 1014 Hz
E = 6.63 × 10-34 × 6 × 1014
E = 3.98 × 10-19 J
2. Power emitted by a laser of frequency 6 × 1014 Hz is 2 × 10-3 W. How many photons per second, on an average, are emitted by the source?
Solution:
To calculate number of photons emitted per second by laser, we use relationship between power (P) and energy (E):
P = N⋅E
where,
- P is Power emitted by the laser (in watts)
- N is Number of photons emitted per second
- E is Energy of a single photon (in joules)
E = hν
Given,
Frequency(v) = 6 × 1014 Hz
E = 6.63 × 10-34 × 6 × 1014
E = 3.98 × 10-19 J
For N:
N = P/E
N = (2×10-3 W)/(3.98×10-19J)
N ≈ 5.025×1015 photons/second
So, on average, approximately 5.025×1015 photons are emitted per second by source.
3. Work function of Cesium is 2.14 eV. Find threshold frequency for Cesium.
Solution:
For Cesium with a work function of 2.14 eV
Φ₀ = 2.14 eV × (1.602176634 × 10-19 J/eV)
Φ₀ ≈ 3.427 × 10-19 J
Now, we equate work function to energy of a photon: Φ₀ = hν₀
Solving for v0:
v0 = Φ₀ /h
Substituting the values:
ν₀ = (3.427 × 10-19 J)/(6.62607015 × 10-34 J·s)
ν₀ ≈ 5.174 × 1014 Hz
So, threshold frequency for Cesium is approximately 5.174 × 1014 Hz.
4. For the above problem, find the wavelength of the incident light if the photocurrent is brought to zero by a stopping potential of 0.6 V.
Solution:
Photoelectric effect equation: eV0 = hν₀ - Φ₀
ν₀ = (eV0 + Φ₀) / h ≌ 6.62 × 1014 Hz
Once we have the frequency, we can use the speed of light formula to find the wavelength:
Æ› = c/ν₀ = 3 × 108 /6.62 × 1014
Æ› = 4.53 × 10-7 m
Æ› = 453 nm
5. Find the frequency of two photons emitted when an electron and a positron collide?
Solution:
Given,
Rest mass of an electron (or positron) is m0= 9.1 × 10-31 kg
Etotal = 2m0c2
Total energy is converted into the energy of the photons produced in the annihilation process. Since two photons are produced, we divide this total energy by two to find the energy of a single photon.
Ephoton = Etotal/2 = m0c2
Now, we can use the equation E = hv to find the frequency of photon:
v = Ephoton/h
v = m0c2/h
v = ( 9.1 × 10-31 × 9 × 1016 )/6.626 x 10-34
v = 1.24 × 1020 Hz
So, frequency of two photons emitted when an electron and a positron annihilate is 1.24 × 1020 Hz.
Practice Problems on Photon
Q1: A hydrogen nucleus (H+) and an antiproton collide and annihilate each other. If the mass of a hydrogen atom (1H) is 1.67×10-27 kg, find the frequency of the two emitted photons.
Q2: If the energy for an electron in the nth orbit around the atom is given by En =-13.6/n2 eV, find the frequency of the photon released when an electron jumps from the 3rd to the 1st orbit.
Q3: Threshold frequency for a certain metal is 3.3 × 1014 Hz. If light of frequency 8.2 × 1014 Hz is incident on the metal, predict the cut-off voltage for the photoelectric emission.
Q4: Light of wavelength 488 nm is produced by an argon laser which is used in the photoelectric effect. When light from this spectral line is incident on the emitter, the stopping potential of photoelectrons is 0.38 V. Find the work function of the material from which the emitter is made.
Q5: Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. Find the energy and momentum of each photon in the light beam.
Photons Frequently Asked Questions
What is a Photon?
A photon is a fundamental particle of electromagnetic radiation, travelling at the speed of light. It is a quantum or discrete energy packet of the electromagnetic field.
What determines energy of a Photon?
Energy of a photon is directly proportional to its frequency, as described by Planck's equation (E = ℎν).
Can Photons Transfer Momentum?
Photons carry momentum (p = h/λ) and can transfer it to other objects through interactions.
Are photons electrons?
No, photons are not electrons. Photon are elementary particle that acts as a carrier of energy, while an electron is a subatomic particle that are responsible for electric property of any material.
Are photons massless?
Rest mass of Photon is zero but as photon is a matter its mass can not be absolute zero. Mass of photon is wave length dependent.
How does a photon travel?
Photons in vacuum, travel at the speed of light.
Wave Nature of Matter and De Broglie’s Equation
One of physics’ most perplexing ideas is the wave nature of matter. A particle is constrained to a certain location, but a wave is dispersed over space. It has been demonstrated that light can have a particle or wave nature. As with a billiard ball, electrons and photons display particle characteristics in the photoelectric effect. However, you’ll recall the Diffraction experiment and the Interference Rings. Similar to the way two waves on a pond’s surface interact as they come together. In many instances, the waveform of light is evident. It’s a fascinating puzzle to solve. It even affects our sense of sight! The eye-lens light collecting and focussing mechanisms are in keeping with light’s wave nature. However, its absorption by the retina’s rods and cones corresponds to light’s particle nature! While we were still trying to figure out the riddle, Louis de Broglie threw a wrench in the works with his de Broglie Relationship.
Wave Nature of Matter
Radiation is viewed as a wave in classical mechanics, while particles are viewed as hard billiard balls. Radiation was shown to be capable of behaving as both waves and particles. Radiation and moving particles may both supply energy and momentum to various things. De Broglie proposed in 1924 that matter should have a dual nature because of nature’s inherent symmetry. Particles do not have a specific location in the space where they reside. Quantum theory was built on the idea that radiation and matter have a dual nature.
De Broglie’s Equation
Light and radiation are both particles and waves, according to De Broglie’s hypothesis, thus matter must also have a particle and wave character. Wave theory was born as a result of the de Broglie connection.
De Broglie’s equation is given as:
λ = h ⁄ p = h ⁄ (mv)
where,
- λ is the wavelength of particle,
- p is the momentum of a particle,
- h is the Planck’s constant,
- m is the mass of particle, &
- v is the velocity of the particle.
Since this connection shows that matter may act like a wave, it’s important to understand its importance. A moving particle, no matter how little or large it may be, has a unique wavelength according to De Broglie’s Equation. If we look closely at macroscopic objects, we can see the wave aspect of the matter. As an item grows in size, its wavelength shrinks until it is undetectable, which explains why macroscopic things in the actual world lack wave-like characteristics. Even the cricket ball you throw has a wavelength that you cannot see. The Plank’s constant links the wavelength and momentum in the equation.
De Broglie’s Hypothesis
The momentum of a photon having energy E is given as:
p = E ⁄ c
The speed of light in a vacuum is represented by the letter c.≥
Planck’s idea states that the energy of a photon is determined by its frequency and wavelength.
E = h v = h c ⁄ λ
The energy should be equal, implying:
h c ⁄ λ = p c
λ = h ⁄ p
De Broglie concluded that the aforementioned relationship should apply to particles as well. p=mv is the momentum of a particle with mass m moving at a speed of v. As a result, it must have a wavelength of
λ = h ⁄ p = h ⁄ (mv)
Heisenberg’s Uncertainty Principle
By diffracting electrons through a crystal, the Davisson-Germer experiment demonstrated without a shadow of a doubt the nature of matter as a wave. De Broglie won the Nobel Prize in Physics in 1929 for his theory of matter waves, which opened up a whole new area of study known as Quantum Physics. Heisenberg’s Uncertainty Principle neatly incorporates the matter-wave hypothesis. According to the Uncertainty Principle, it is impossible to know an electron’s momentum and position at the same time for any other particle. Uncertainty exists in both the position ‘Δx’ and momentum ‘Δp’.
Heisenberg’s Uncertainty Equation:
The Uncertainty Principle says that a particle’s momentum and location cannot be determined with precision at the same time. To put it another way, there is always some degree of ambiguity about where something is located, as well as its velocity. The unknowns are linked by,
Δx Δp ≥ h ⁄ 4Ï€
where
- Δx is the uncertainty in position and
- Δp is the uncertainty in the momentum of the particle.
If a particle’s momentum is precisely measured (i.e. Δp=0), the uncertainty Δx in its location becomes infinite. According to de Broglie’s equation, a particle with a known momentum should also have a known wavelength. A certain wavelength can be found throughout all of space, all the way to infinity. According to Born’s Probability Interpretation, this implies that the particle is not localised in space and that the uncertainty of its location is thus limitless. The wavelengths in real life, on the other hand, have a defined boundary and aren’t limitless, thus uncertainty in terms of both location and momentum is limited. Localized waves (wave packets), which include wavelengths of varying lengths, should be used to represent any particle.
Sample Problems
Question 1: What do you understand by the Matter-wave packet?
Answer:
In contrast to a progressive wave, a wave packet is a superposition of sinusoidal waves of various wavelengths that is confined in space. The location and momentum of a particle may be accurately represented using a wave packet. The particle’s velocity is calculated using the packet’s group velocity. The De Broglie hypothesis and the uncertainty principle are both used to describe a wave packet.
Question 2: Can the De Broglie Equation Be Used to Calculate Photon Energy?
Answer:
Radiation is made up of photons, which are massless particles. Even though a photon’s rest mass is zero, relativity says its energy equates to a momentum. A photon’s energy is related to its frequency and wavelength, according to Max Planck’s theory. The relationship between wavelength and photon momentum resembles that of the de Broglie equation for matter.
Question 3: If a baseball weighs 0.1 kg and travels at 60 m ⁄ s, what is its de Broglie wavelength?
Answer:
Given:
Mass of a baseball, m = 0.1 kg
Speed of a baseball, v = 60 m ⁄ s
Planck’s constant, h = 6.626 × 10−34 J s
The de Broglie wavelength of an object is given as:
λ = h ⁄ (m v)
= 6.626 × 10−34 ⁄ ( 0.1 × 60) m
= 1.104 × 10−34 m
Hence, the de Broglie wavelength is 1.104 × 10−34 m.
Question 4: Which has a longer de Broglie wavelength if electron and proton are the same speed?
Answer:
Due to its far greater mass, the de Broglie wavelength of a proton is 1800 times smaller than that of an electron, and as a result, its momentum at the same speed is 1800 times more than that of an electron. The electron has a greater radiance because of its longer wavelength.
Question 5: A molecule’s electron moves at a 20 m ⁄ s clip. The electron’s p-momentum uncertainty is 2p×10−6 that of the electron’s initial momentum. Calculate the uncertainty in position x for an electron weighing 9.1×10−31 kg.
Answer:
Given:
Mass of the electron, m = 9.1×10−31 kg
Speed of the electron, v = 20 m ⁄ s
Momentum of the electron, p = mv
= 9.1×10−31 × 20 kg m ⁄ s
= 182×10−31 kg m/s
Uncertainty in momentum, Δp = 2p×10−6
= 364×10−37 kg m/s
Planck’s constant, h = 6.626 × 10−34 J s
Heisenberg Uncertainty Formula is given as:
Δx Δp ≥ h ⁄ 4Ï€
Δx ≥ h ⁄ (4Ï€ Δp)
Δx ≥ (6.626 × 10−34 J s) ⁄ (4Ï€ × 364×10−37 kg m/s) = 1.44 m
Hence, the uncertainty in position for an electron is 1.44 m.
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