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# PHOTONS, MODERN PHYSICS JEE Notes | EduRev

## JEE : PHOTONS, MODERN PHYSICS JEE Notes | EduRev

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Module 5 : MODERN PHYSICS
Lecture 24 : PHOTONS

Objectives

In this lecture you will learn the following
Radiation itself is quantized and consists of a collection of particles called photons.
The phenomenon of photoelectric effect and its characteristics.
How classical wave theory of radiation fails in explaining photoelectric effect.
Einstein's theory of photoelectric effect.

PHOTONS

Planck's explanation of black body radiation was revolutionary as it suggested that atoms could exchange
energy only in multiples of quantum of energy. Five years later, in 1905, Einstein put forward a theory of
photoelectric effect which suggested that the quantum of energy was not a property associated with the
consists of discrete bundles of energy, called photons. Thus, electromagnetic energy is seen as a collection
of photons. A photon is characterized by an energy , related to the frequency by the relationship

Further, each photon carries a momentum  given by

In Einstein's special theory of relativity, the energy of a particle of rest mass  and momentum  is given
by

which implies that photons have zero rest mass.

Exercise 1

Calculate the wavelength of a 2eV photon. (  J.)
(Ans. 620 nm)

Example-6

Earth receives 1.4 kW of energy from the sun. If it is assumed that the sunlight consists of monochromatic
radiation of wavelength 600 nm, how many photons arrive at the earth every second ?

Solution

If  nm, the corresponding frequency is  Hz. Thus the number of photons
per second is
Page 2

Module 5 : MODERN PHYSICS
Lecture 24 : PHOTONS

Objectives

In this lecture you will learn the following
Radiation itself is quantized and consists of a collection of particles called photons.
The phenomenon of photoelectric effect and its characteristics.
How classical wave theory of radiation fails in explaining photoelectric effect.
Einstein's theory of photoelectric effect.

PHOTONS

Planck's explanation of black body radiation was revolutionary as it suggested that atoms could exchange
energy only in multiples of quantum of energy. Five years later, in 1905, Einstein put forward a theory of
photoelectric effect which suggested that the quantum of energy was not a property associated with the
consists of discrete bundles of energy, called photons. Thus, electromagnetic energy is seen as a collection
of photons. A photon is characterized by an energy , related to the frequency by the relationship

Further, each photon carries a momentum  given by

In Einstein's special theory of relativity, the energy of a particle of rest mass  and momentum  is given
by

which implies that photons have zero rest mass.

Exercise 1

Calculate the wavelength of a 2eV photon. (  J.)
(Ans. 620 nm)

Example-6

Earth receives 1.4 kW of energy from the sun. If it is assumed that the sunlight consists of monochromatic
radiation of wavelength 600 nm, how many photons arrive at the earth every second ?

Solution

If  nm, the corresponding frequency is  Hz. Thus the number of photons
per second is

Example-7

Assuming the formula for black body radiation to the valid for the universe, calculate the number density of
photons in the universe due to cosmic microwave background.

Solution

Taking the expression for energy density in the interval  and

the number density of photons with energy in this frequency interval is obtained by dividing the above
expression by . The total number density of photons is obtained by integrating the above expression
over all frequencies

Substitute ,

The integral has to be done numerically, say by using Simpson's rule. The value of the integral is 2.4, which
gives the number density of photons of the cosmic radiation to be  per m .

Exercise 2

Assuming the sun to be a black body, calculate the number of photons emitted by the sun every second.
(Ans. )
Photoelectric Effect

When light falls on certain metals, electrons are ejected from the surface of the metal. In the arrangement
shown in the figure, the wire marked anode is held at positive potential with respect to the curved plate
marked cathode

When light of certain minimum frequency falls on the cathode, electrons are emitted in all directions. These
electrons are called photoelectrons . Some of these electrons reach the anode wire which provides a path
to the electrons to give rise to a mesurable photo-current . By making the anode more positive with
respect to the cathode, more electrons are attracted towards the anode and the photo-current increases.
When the anode potential is such that all the emitted electrons reach the anode, any further increase in the
anode voltage does not increase current any further.
Page 3

Module 5 : MODERN PHYSICS
Lecture 24 : PHOTONS

Objectives

In this lecture you will learn the following
Radiation itself is quantized and consists of a collection of particles called photons.
The phenomenon of photoelectric effect and its characteristics.
How classical wave theory of radiation fails in explaining photoelectric effect.
Einstein's theory of photoelectric effect.

PHOTONS

Planck's explanation of black body radiation was revolutionary as it suggested that atoms could exchange
energy only in multiples of quantum of energy. Five years later, in 1905, Einstein put forward a theory of
photoelectric effect which suggested that the quantum of energy was not a property associated with the
consists of discrete bundles of energy, called photons. Thus, electromagnetic energy is seen as a collection
of photons. A photon is characterized by an energy , related to the frequency by the relationship

Further, each photon carries a momentum  given by

In Einstein's special theory of relativity, the energy of a particle of rest mass  and momentum  is given
by

which implies that photons have zero rest mass.

Exercise 1

Calculate the wavelength of a 2eV photon. (  J.)
(Ans. 620 nm)

Example-6

Earth receives 1.4 kW of energy from the sun. If it is assumed that the sunlight consists of monochromatic
radiation of wavelength 600 nm, how many photons arrive at the earth every second ?

Solution

If  nm, the corresponding frequency is  Hz. Thus the number of photons
per second is

Example-7

Assuming the formula for black body radiation to the valid for the universe, calculate the number density of
photons in the universe due to cosmic microwave background.

Solution

Taking the expression for energy density in the interval  and

the number density of photons with energy in this frequency interval is obtained by dividing the above
expression by . The total number density of photons is obtained by integrating the above expression
over all frequencies

Substitute ,

The integral has to be done numerically, say by using Simpson's rule. The value of the integral is 2.4, which
gives the number density of photons of the cosmic radiation to be  per m .

Exercise 2

Assuming the sun to be a black body, calculate the number of photons emitted by the sun every second.
(Ans. )
Photoelectric Effect

When light falls on certain metals, electrons are ejected from the surface of the metal. In the arrangement
shown in the figure, the wire marked anode is held at positive potential with respect to the curved plate
marked cathode

When light of certain minimum frequency falls on the cathode, electrons are emitted in all directions. These
electrons are called photoelectrons . Some of these electrons reach the anode wire which provides a path
to the electrons to give rise to a mesurable photo-current . By making the anode more positive with
respect to the cathode, more electrons are attracted towards the anode and the photo-current increases.
When the anode potential is such that all the emitted electrons reach the anode, any further increase in the
anode voltage does not increase current any further.

See the animation

If the voltage is reversed making the anode negative with respect to the cathode, the electrons get
decelerated and only the more energetic of the electrons can reach the anode.

If the reverse voltage is such that even the electrons which are ejected with the maximum kinetic energy
cannot overcome the potential, the photo-current becomes zero. The reverse voltage which is just enough to
stop the most energetic photoelectrons is called the stopping potential. If  is the maximum kinetic
energy of the electrons, the stopping potential is  defined by

The photoelectric effect exhibits the following features :
Page 4

Module 5 : MODERN PHYSICS
Lecture 24 : PHOTONS

Objectives

In this lecture you will learn the following
Radiation itself is quantized and consists of a collection of particles called photons.
The phenomenon of photoelectric effect and its characteristics.
How classical wave theory of radiation fails in explaining photoelectric effect.
Einstein's theory of photoelectric effect.

PHOTONS

Planck's explanation of black body radiation was revolutionary as it suggested that atoms could exchange
energy only in multiples of quantum of energy. Five years later, in 1905, Einstein put forward a theory of
photoelectric effect which suggested that the quantum of energy was not a property associated with the
consists of discrete bundles of energy, called photons. Thus, electromagnetic energy is seen as a collection
of photons. A photon is characterized by an energy , related to the frequency by the relationship

Further, each photon carries a momentum  given by

In Einstein's special theory of relativity, the energy of a particle of rest mass  and momentum  is given
by

which implies that photons have zero rest mass.

Exercise 1

Calculate the wavelength of a 2eV photon. (  J.)
(Ans. 620 nm)

Example-6

Earth receives 1.4 kW of energy from the sun. If it is assumed that the sunlight consists of monochromatic
radiation of wavelength 600 nm, how many photons arrive at the earth every second ?

Solution

If  nm, the corresponding frequency is  Hz. Thus the number of photons
per second is

Example-7

Assuming the formula for black body radiation to the valid for the universe, calculate the number density of
photons in the universe due to cosmic microwave background.

Solution

Taking the expression for energy density in the interval  and

the number density of photons with energy in this frequency interval is obtained by dividing the above
expression by . The total number density of photons is obtained by integrating the above expression
over all frequencies

Substitute ,

The integral has to be done numerically, say by using Simpson's rule. The value of the integral is 2.4, which
gives the number density of photons of the cosmic radiation to be  per m .

Exercise 2

Assuming the sun to be a black body, calculate the number of photons emitted by the sun every second.
(Ans. )
Photoelectric Effect

When light falls on certain metals, electrons are ejected from the surface of the metal. In the arrangement
shown in the figure, the wire marked anode is held at positive potential with respect to the curved plate
marked cathode

When light of certain minimum frequency falls on the cathode, electrons are emitted in all directions. These
electrons are called photoelectrons . Some of these electrons reach the anode wire which provides a path
to the electrons to give rise to a mesurable photo-current . By making the anode more positive with
respect to the cathode, more electrons are attracted towards the anode and the photo-current increases.
When the anode potential is such that all the emitted electrons reach the anode, any further increase in the
anode voltage does not increase current any further.

See the animation

If the voltage is reversed making the anode negative with respect to the cathode, the electrons get
decelerated and only the more energetic of the electrons can reach the anode.

If the reverse voltage is such that even the electrons which are ejected with the maximum kinetic energy
cannot overcome the potential, the photo-current becomes zero. The reverse voltage which is just enough to
stop the most energetic photoelectrons is called the stopping potential. If  is the maximum kinetic
energy of the electrons, the stopping potential is  defined by

The photoelectric effect exhibits the following features :
Photoelectrons are not ejected unless the frequency of incident light is above a certain threshold value

. The value of  depends on the material of the cathode.
If the frequency of incident radiation  is greater than , even a light of very weak intensity will cause

photoelectrons to be emitted. If , even the most intense light will not cause photoelectrons to be
emitted. For  , the photo-current increases linearly with the intensity of light.

The maximum kinetic energy of the photoelectrons depend on the frequency of incident radiation and not on
its intensity.
The emission of photoelectrons is almost instantaneous. i.e. there is no time lag between the emission of
electrons and switching on of the light source.
Failure of Classical Wave Theory
According to the classical wave theory, when electromagnetic wave falls on the surface of a metal, an atom
on the surface will absorb energy from the electric field of the wave. The rate at which the energy is
absorbed depends on the surface area of the atom. An electron can be dislodged from an atom once it
absorbes sufficient amount of energy. By increasing the intensity of light (irrespective of its frequency) more
energy can be transferred to the atom causing electrons to be ejected.

What is observed is that unless  photoelectrons are not emitted, no matter how intense the
Further, according to wave theory, the kinetic energy of emitted electrons would increase with the intensity
of light as it would impart more energy to an electron. However, the kinetic energy of photoelectrons is
found to depend only on the frequency of radiation and not on the intensity.

Page 5

Module 5 : MODERN PHYSICS
Lecture 24 : PHOTONS

Objectives

In this lecture you will learn the following
Radiation itself is quantized and consists of a collection of particles called photons.
The phenomenon of photoelectric effect and its characteristics.
How classical wave theory of radiation fails in explaining photoelectric effect.
Einstein's theory of photoelectric effect.

PHOTONS

Planck's explanation of black body radiation was revolutionary as it suggested that atoms could exchange
energy only in multiples of quantum of energy. Five years later, in 1905, Einstein put forward a theory of
photoelectric effect which suggested that the quantum of energy was not a property associated with the
consists of discrete bundles of energy, called photons. Thus, electromagnetic energy is seen as a collection
of photons. A photon is characterized by an energy , related to the frequency by the relationship

Further, each photon carries a momentum  given by

In Einstein's special theory of relativity, the energy of a particle of rest mass  and momentum  is given
by

which implies that photons have zero rest mass.

Exercise 1

Calculate the wavelength of a 2eV photon. (  J.)
(Ans. 620 nm)

Example-6

Earth receives 1.4 kW of energy from the sun. If it is assumed that the sunlight consists of monochromatic
radiation of wavelength 600 nm, how many photons arrive at the earth every second ?

Solution

If  nm, the corresponding frequency is  Hz. Thus the number of photons
per second is

Example-7

Assuming the formula for black body radiation to the valid for the universe, calculate the number density of
photons in the universe due to cosmic microwave background.

Solution

Taking the expression for energy density in the interval  and

the number density of photons with energy in this frequency interval is obtained by dividing the above
expression by . The total number density of photons is obtained by integrating the above expression
over all frequencies

Substitute ,

The integral has to be done numerically, say by using Simpson's rule. The value of the integral is 2.4, which
gives the number density of photons of the cosmic radiation to be  per m .

Exercise 2

Assuming the sun to be a black body, calculate the number of photons emitted by the sun every second.
(Ans. )
Photoelectric Effect

When light falls on certain metals, electrons are ejected from the surface of the metal. In the arrangement
shown in the figure, the wire marked anode is held at positive potential with respect to the curved plate
marked cathode

When light of certain minimum frequency falls on the cathode, electrons are emitted in all directions. These
electrons are called photoelectrons . Some of these electrons reach the anode wire which provides a path
to the electrons to give rise to a mesurable photo-current . By making the anode more positive with
respect to the cathode, more electrons are attracted towards the anode and the photo-current increases.
When the anode potential is such that all the emitted electrons reach the anode, any further increase in the
anode voltage does not increase current any further.

See the animation

If the voltage is reversed making the anode negative with respect to the cathode, the electrons get
decelerated and only the more energetic of the electrons can reach the anode.

If the reverse voltage is such that even the electrons which are ejected with the maximum kinetic energy
cannot overcome the potential, the photo-current becomes zero. The reverse voltage which is just enough to
stop the most energetic photoelectrons is called the stopping potential. If  is the maximum kinetic
energy of the electrons, the stopping potential is  defined by

The photoelectric effect exhibits the following features :
Photoelectrons are not ejected unless the frequency of incident light is above a certain threshold value

. The value of  depends on the material of the cathode.
If the frequency of incident radiation  is greater than , even a light of very weak intensity will cause

photoelectrons to be emitted. If , even the most intense light will not cause photoelectrons to be
emitted. For  , the photo-current increases linearly with the intensity of light.

The maximum kinetic energy of the photoelectrons depend on the frequency of incident radiation and not on
its intensity.
The emission of photoelectrons is almost instantaneous. i.e. there is no time lag between the emission of
electrons and switching on of the light source.
Failure of Classical Wave Theory
According to the classical wave theory, when electromagnetic wave falls on the surface of a metal, an atom
on the surface will absorb energy from the electric field of the wave. The rate at which the energy is
absorbed depends on the surface area of the atom. An electron can be dislodged from an atom once it
absorbes sufficient amount of energy. By increasing the intensity of light (irrespective of its frequency) more
energy can be transferred to the atom causing electrons to be ejected.

What is observed is that unless  photoelectrons are not emitted, no matter how intense the
Further, according to wave theory, the kinetic energy of emitted electrons would increase with the intensity
of light as it would impart more energy to an electron. However, the kinetic energy of photoelectrons is
found to depend only on the frequency of radiation and not on the intensity.

Another problem with the classical theory is that it would predict a time lag between the time light falls on a
surface and the instant photoelectrons are emitted. The reason why one would expect such a time lag is that
the surface are of an atom is very small, as a result of which an atom can only absorb a small fraction of
energy that falls on the surface. The following example gives a rough estimate of the expected time lag.
However, it is observed that the emission of electrons is practically instantaneous, with time lag, if
any, being less than  seconds.

Example-8

Consider a light source such as a laser with a power output of 1mW spread over a narrow beam of cross
section 0.1 cm  falling on a surface of a metal. Estimate the time lag of photoelectron emission as per wave
theory.

Solution

Taking atomic diameter to be of the order of  cm, the area exposed to the beam is  cm .
The fraction of light energy absorbed is . Thus the energy absorbed from the beam
every second is  J, which is approximately 6 eV (1 eV =  J). The
amount of energy required to ionize an atom by dislodging an electron is typically 10 eV. Thus it takes about
1.6 seconds to absorb the required energy which is rough estimate of the time lag.

Einstein's Photoelectric Equation :

According to Einstein's explanation, photoelectric effect occurs due to absorption of a single photon by an
electron in the atom. When radiation falls on a metal surface, an electron may absorb one quantum of
energy and increase its energy by . Some of the absorbed energy, W, will be used to separate the
electron from the metal surface. The surplus energy appears as the kinetic energy of the emitted electron

The electrons which are more tightly bound to the metal (e.g. electrons which lie two or three atomic layers
below the surface) require more energy to be removed. We define Work Function  of a metal as the
minimu energy that must be supplied to an electron at the metal surface to dislodge it from the metal. Such
electrons are emitted with maximum possible kinetic energy. Thus Einstein's equation becomes

Since kinetic energy cannot be negative, the above equation implies the existence of a minimum frequency
for photoemission to take place

Using this, we can reqrite rewrite Einstein's equation as

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