Notes - Modern Physics Notes | EduRev

: Notes - Modern Physics Notes | EduRev

 Page 1


Modern Physics
V H Satheeshkumar
Department of Physics
Sri Bhagawan Mahaveer Jain College of Engineering
Jain Global Campus, Kanakapura Road
Bangalore 562 112, India.
vhsatheeshkumar@gmail.com
October 1, 2008
Who can use this?
The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the
?rst of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for
the ?rst-semester (September 2008 - January 2009) BE students of all branches. Any
student interested in exploring more about the course may visit the course homepage
at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of
studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-)
Syllabus as prescribed by VTU
Introduction to blackbody radiation spectrum; Photoelectric effect;
Compton effect; Wave particle Dualism; de Broglie hypothesis:de Broglie
wavelength, Extension to electron particle; Davisson and Germer
Experiment, Matter waves and their Characteristic properties; Phase
velocity, group velocity and particle velocity; Relation between phase
velocity and group velocity; Relation between group velocity and
particle velocity; Expression for de Broglie wavelength using group
velocity.
Reference
• Arthur Beiser, Concepts of Modern Physics, 6
th
Edition, Tata McGraw-Hill Pub-
lishing Company Limited, ISBN- 0-07-049553-X.
————————————
This document is typeset in Free Software L
A
T
E
X2e distributed under the terms of the GNU General Public License.
1
Page 2


Modern Physics
V H Satheeshkumar
Department of Physics
Sri Bhagawan Mahaveer Jain College of Engineering
Jain Global Campus, Kanakapura Road
Bangalore 562 112, India.
vhsatheeshkumar@gmail.com
October 1, 2008
Who can use this?
The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the
?rst of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for
the ?rst-semester (September 2008 - January 2009) BE students of all branches. Any
student interested in exploring more about the course may visit the course homepage
at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of
studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-)
Syllabus as prescribed by VTU
Introduction to blackbody radiation spectrum; Photoelectric effect;
Compton effect; Wave particle Dualism; de Broglie hypothesis:de Broglie
wavelength, Extension to electron particle; Davisson and Germer
Experiment, Matter waves and their Characteristic properties; Phase
velocity, group velocity and particle velocity; Relation between phase
velocity and group velocity; Relation between group velocity and
particle velocity; Expression for de Broglie wavelength using group
velocity.
Reference
• Arthur Beiser, Concepts of Modern Physics, 6
th
Edition, Tata McGraw-Hill Pub-
lishing Company Limited, ISBN- 0-07-049553-X.
————————————
This document is typeset in Free Software L
A
T
E
X2e distributed under the terms of the GNU General Public License.
1
www.satheesh.bigbig.com/EnggPhy 2
1 Introduction
Theturnofthe20
th
centurybroughtthestartofarevolutioninphysics. In1900,MaxPlanckpublished
his explanation of blackbody radiation. This equation assumed that radiators are quantized, which
proved to be the opening argument in the edi?ce that would become quantum mechanics. In this
chapter, many of the developments which form the foundation of modern physics are discussed.
2 Blackbody radiation spectrum
A blackbody is an object that absorbs all light that falls on it. Since no light is re?ected or transmitted,
the object appears black when it is cold. The term blackbody was introduced by Gustav Kirchho? in
1860. A perfect blackbody, in thermal equilibrium, will emit exactly as much as it absorbs at every
wavelength. The light emitted by a blackbody is called blackbody radiation.
Theplotofdistributionofemittedenergyasafunctionofwavelengthandtemperatureofblackbody
is know as blackbody spectrum. It has the following characteristics.
• The spectral distribution of energy in the radiation depends only on the temperature of the
blackbody.
• The higher the temperature, the greater the amount of total radiation energy emitted and also
energy emitted at individual wavelengths.
• The higher the temperature, the lower the wavelength at which maximum emission occurs.
Many theories were proposed to explain the nature of blackbody radiation based on classical physics
arguments. But non of them could explain the complete blackbody spectrum satisfactorily. These
theories are discussed in brief below.
Page 3


Modern Physics
V H Satheeshkumar
Department of Physics
Sri Bhagawan Mahaveer Jain College of Engineering
Jain Global Campus, Kanakapura Road
Bangalore 562 112, India.
vhsatheeshkumar@gmail.com
October 1, 2008
Who can use this?
The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the
?rst of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for
the ?rst-semester (September 2008 - January 2009) BE students of all branches. Any
student interested in exploring more about the course may visit the course homepage
at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of
studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-)
Syllabus as prescribed by VTU
Introduction to blackbody radiation spectrum; Photoelectric effect;
Compton effect; Wave particle Dualism; de Broglie hypothesis:de Broglie
wavelength, Extension to electron particle; Davisson and Germer
Experiment, Matter waves and their Characteristic properties; Phase
velocity, group velocity and particle velocity; Relation between phase
velocity and group velocity; Relation between group velocity and
particle velocity; Expression for de Broglie wavelength using group
velocity.
Reference
• Arthur Beiser, Concepts of Modern Physics, 6
th
Edition, Tata McGraw-Hill Pub-
lishing Company Limited, ISBN- 0-07-049553-X.
————————————
This document is typeset in Free Software L
A
T
E
X2e distributed under the terms of the GNU General Public License.
1
www.satheesh.bigbig.com/EnggPhy 2
1 Introduction
Theturnofthe20
th
centurybroughtthestartofarevolutioninphysics. In1900,MaxPlanckpublished
his explanation of blackbody radiation. This equation assumed that radiators are quantized, which
proved to be the opening argument in the edi?ce that would become quantum mechanics. In this
chapter, many of the developments which form the foundation of modern physics are discussed.
2 Blackbody radiation spectrum
A blackbody is an object that absorbs all light that falls on it. Since no light is re?ected or transmitted,
the object appears black when it is cold. The term blackbody was introduced by Gustav Kirchho? in
1860. A perfect blackbody, in thermal equilibrium, will emit exactly as much as it absorbs at every
wavelength. The light emitted by a blackbody is called blackbody radiation.
Theplotofdistributionofemittedenergyasafunctionofwavelengthandtemperatureofblackbody
is know as blackbody spectrum. It has the following characteristics.
• The spectral distribution of energy in the radiation depends only on the temperature of the
blackbody.
• The higher the temperature, the greater the amount of total radiation energy emitted and also
energy emitted at individual wavelengths.
• The higher the temperature, the lower the wavelength at which maximum emission occurs.
Many theories were proposed to explain the nature of blackbody radiation based on classical physics
arguments. But non of them could explain the complete blackbody spectrum satisfactorily. These
theories are discussed in brief below.
www.satheesh.bigbig.com/EnggPhy 3
2.1 Stefan-Boltzmann law
The Stefan-Boltzmann law, also known as Stefan’s law, states that the total energy radiated per unit
surface area of a blackbody in unit time (known variously as the blackbody irradiance, energy ?ux
density, radiant ?ux, or the emissive power), E
?
, is directly proportional to the fourth power of the
blackbody’s thermodynamic temperature T (also called absolute temperature):
E
?
= sT
4
. (1)
The constant of proportionality s is called the Stefan-Boltzmann constant or Stefan’s constant. It is
not a fundamental constant, in the sense that it can be derived from other known constants of nature.
The value of the constant is 5.6704×10
-8
Js
-1
m
-2
K
-4
. The Stefan-Boltzmann law is an example of
a power law.
2.2 Wien’s displacement law
Wien’s displacement law states that there is an inverse relationship between the wavelength of the peak
of the emission of a blackbody and its absolute temperature.
?
max
?
1
T
T?
max
= b (2)
where
?
max
is the peak wavelength in meters,
T is the temperature of the blackbody in kelvins (K), and
bisaconstantofproportionality, calledWien’sdisplacementconstantandequals2.8978×10
-3
mK.
In other words, Wien’s displacement law states that the hotter an object is, the shorter the wavelength
at which it will emit most of its radiation.
2.3 Wien’s distribution law
According to Wein, the energy density, E
?
, emitted by a blackbody in a wavelength interval ? and
?+d? is given by
E
?
d? =
c
1
?
5
e
(-c
2
/?T)
d? (3)
where c
1
and c
2
are constants. This is known as Wien’s distribution law or simply Wein’s law. This
law holds good for smaller values of ? but does not match the experimental results for larger values of
?. Wien received the 1911 Nobel Prize for his work on heat radiation.
2.4 Rayleigh-Jeans’ law
According to Rayleigh and Jeans the energy density, E
?
, emitted by a blackbody in a wavelength
interval ? and ?+d? is given by
E
?
d? =
8pkT
?
4
d?, (4)
where k is the Boltzmann’s constant whose value is equal to 1.381×10
-23
JK
-1
.
It agrees well with experimental measurements for long wavelengths. However it predicts an en-
ergy output that diverges towards in?nity as wavelengths grow smaller. This was not supported by
experiments and the failure has become known as the ultraviolet catastrophe or Rayleigh-Jeans
catastrophe. Here the word ultraviolet signi?es shorter wavelength or higher frequencies and not the
ultraviolet region of the spectrum. One more thing to note is that, it was not, as is sometimes asserted
in physics textbooks, a motivation for quantum theory.
Page 4


Modern Physics
V H Satheeshkumar
Department of Physics
Sri Bhagawan Mahaveer Jain College of Engineering
Jain Global Campus, Kanakapura Road
Bangalore 562 112, India.
vhsatheeshkumar@gmail.com
October 1, 2008
Who can use this?
The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the
?rst of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for
the ?rst-semester (September 2008 - January 2009) BE students of all branches. Any
student interested in exploring more about the course may visit the course homepage
at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of
studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-)
Syllabus as prescribed by VTU
Introduction to blackbody radiation spectrum; Photoelectric effect;
Compton effect; Wave particle Dualism; de Broglie hypothesis:de Broglie
wavelength, Extension to electron particle; Davisson and Germer
Experiment, Matter waves and their Characteristic properties; Phase
velocity, group velocity and particle velocity; Relation between phase
velocity and group velocity; Relation between group velocity and
particle velocity; Expression for de Broglie wavelength using group
velocity.
Reference
• Arthur Beiser, Concepts of Modern Physics, 6
th
Edition, Tata McGraw-Hill Pub-
lishing Company Limited, ISBN- 0-07-049553-X.
————————————
This document is typeset in Free Software L
A
T
E
X2e distributed under the terms of the GNU General Public License.
1
www.satheesh.bigbig.com/EnggPhy 2
1 Introduction
Theturnofthe20
th
centurybroughtthestartofarevolutioninphysics. In1900,MaxPlanckpublished
his explanation of blackbody radiation. This equation assumed that radiators are quantized, which
proved to be the opening argument in the edi?ce that would become quantum mechanics. In this
chapter, many of the developments which form the foundation of modern physics are discussed.
2 Blackbody radiation spectrum
A blackbody is an object that absorbs all light that falls on it. Since no light is re?ected or transmitted,
the object appears black when it is cold. The term blackbody was introduced by Gustav Kirchho? in
1860. A perfect blackbody, in thermal equilibrium, will emit exactly as much as it absorbs at every
wavelength. The light emitted by a blackbody is called blackbody radiation.
Theplotofdistributionofemittedenergyasafunctionofwavelengthandtemperatureofblackbody
is know as blackbody spectrum. It has the following characteristics.
• The spectral distribution of energy in the radiation depends only on the temperature of the
blackbody.
• The higher the temperature, the greater the amount of total radiation energy emitted and also
energy emitted at individual wavelengths.
• The higher the temperature, the lower the wavelength at which maximum emission occurs.
Many theories were proposed to explain the nature of blackbody radiation based on classical physics
arguments. But non of them could explain the complete blackbody spectrum satisfactorily. These
theories are discussed in brief below.
www.satheesh.bigbig.com/EnggPhy 3
2.1 Stefan-Boltzmann law
The Stefan-Boltzmann law, also known as Stefan’s law, states that the total energy radiated per unit
surface area of a blackbody in unit time (known variously as the blackbody irradiance, energy ?ux
density, radiant ?ux, or the emissive power), E
?
, is directly proportional to the fourth power of the
blackbody’s thermodynamic temperature T (also called absolute temperature):
E
?
= sT
4
. (1)
The constant of proportionality s is called the Stefan-Boltzmann constant or Stefan’s constant. It is
not a fundamental constant, in the sense that it can be derived from other known constants of nature.
The value of the constant is 5.6704×10
-8
Js
-1
m
-2
K
-4
. The Stefan-Boltzmann law is an example of
a power law.
2.2 Wien’s displacement law
Wien’s displacement law states that there is an inverse relationship between the wavelength of the peak
of the emission of a blackbody and its absolute temperature.
?
max
?
1
T
T?
max
= b (2)
where
?
max
is the peak wavelength in meters,
T is the temperature of the blackbody in kelvins (K), and
bisaconstantofproportionality, calledWien’sdisplacementconstantandequals2.8978×10
-3
mK.
In other words, Wien’s displacement law states that the hotter an object is, the shorter the wavelength
at which it will emit most of its radiation.
2.3 Wien’s distribution law
According to Wein, the energy density, E
?
, emitted by a blackbody in a wavelength interval ? and
?+d? is given by
E
?
d? =
c
1
?
5
e
(-c
2
/?T)
d? (3)
where c
1
and c
2
are constants. This is known as Wien’s distribution law or simply Wein’s law. This
law holds good for smaller values of ? but does not match the experimental results for larger values of
?. Wien received the 1911 Nobel Prize for his work on heat radiation.
2.4 Rayleigh-Jeans’ law
According to Rayleigh and Jeans the energy density, E
?
, emitted by a blackbody in a wavelength
interval ? and ?+d? is given by
E
?
d? =
8pkT
?
4
d?, (4)
where k is the Boltzmann’s constant whose value is equal to 1.381×10
-23
JK
-1
.
It agrees well with experimental measurements for long wavelengths. However it predicts an en-
ergy output that diverges towards in?nity as wavelengths grow smaller. This was not supported by
experiments and the failure has become known as the ultraviolet catastrophe or Rayleigh-Jeans
catastrophe. Here the word ultraviolet signi?es shorter wavelength or higher frequencies and not the
ultraviolet region of the spectrum. One more thing to note is that, it was not, as is sometimes asserted
in physics textbooks, a motivation for quantum theory.
www.satheesh.bigbig.com/EnggPhy 4
2.5 Planck’s law of black-body radiation
Explaining the blackbody radiation curve was a major challenge in theoretical physics during the late
nineteenth century. All the theories based on classical ideas failed in one or the other way. The
wavelength at which the radiation is strongest is given by Wien’s displacement law, and the overall
power emitted per unit area is given by the Stefan-Boltzmann law. Wein’s law could explain the
blackbody radiation curve only for shorter wavelengths whereas Rayleigh-Jeans’ law worked well only
for larger wavelengths. The problem was ?nally solved in 1901 by Max Planck.
Planck came up with the following formula for the spectral energy density of blackbody radiation
in a wavelength range ? and ?+d?,
E
?
d? =
8phc
?
5
1
e
hc/?kT
-1
d?, (5)
where h is the Planck’s constant whose value is 6.626×10
-34
Js. This formula could explain the entire
blackbody spectrum and does not su?er from an ultraviolet catastrophe unlike the previous ones. But
the problem was to justify it in terms of physical principles. Planck proposed a radically new idea that
the oscillators in the blackbody do not have continuous distribution of energies but only in discrete
amounts. An oscillator emits radiation of frequency ? when it drops from one energy state to the next
lower one, and it jumps to the next higher state when it absorbs radiation of frequency ?. Each such
discrete bundle of energy h? is called quantum. Hence, the energy of an oscillator can be written as
E
n
= nh? n = 0,1,2,3,.... (6)
2.5.1 Derivation Wien’s law from Planck’s law
The Planck’s law of blackbody radiation expressed in terms of wavelength is given by
E
?
d? =
8phc
?
5
1
e
hc/?kT
-1
d?.
In the limit of shorter wavelengths, hc/?kT becomes very small resulting in
e
hc/?kT
 1.
Therefore
e
hc/?kT
-1˜ e
hc/?kT
.
This reduces the Planck’s law to
E
?
d? =
8phc
?
5
1
e
hc/?kT
d?
or
E
?
d? =
8phc
?
5
e
-hc/?kT
d?.
Now identifying 8phc as c
1
and hc/k as c
2
, the above equation takes the form
E
?
d? =
c
1
?
5
e
(-c
2
/?T)
d?.
This is the familiar Wien’s law.
Page 5


Modern Physics
V H Satheeshkumar
Department of Physics
Sri Bhagawan Mahaveer Jain College of Engineering
Jain Global Campus, Kanakapura Road
Bangalore 562 112, India.
vhsatheeshkumar@gmail.com
October 1, 2008
Who can use this?
The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the
?rst of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for
the ?rst-semester (September 2008 - January 2009) BE students of all branches. Any
student interested in exploring more about the course may visit the course homepage
at www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy of
studying this: this chapter is worth 20 marks in the ?nal exam! Cheers ;-)
Syllabus as prescribed by VTU
Introduction to blackbody radiation spectrum; Photoelectric effect;
Compton effect; Wave particle Dualism; de Broglie hypothesis:de Broglie
wavelength, Extension to electron particle; Davisson and Germer
Experiment, Matter waves and their Characteristic properties; Phase
velocity, group velocity and particle velocity; Relation between phase
velocity and group velocity; Relation between group velocity and
particle velocity; Expression for de Broglie wavelength using group
velocity.
Reference
• Arthur Beiser, Concepts of Modern Physics, 6
th
Edition, Tata McGraw-Hill Pub-
lishing Company Limited, ISBN- 0-07-049553-X.
————————————
This document is typeset in Free Software L
A
T
E
X2e distributed under the terms of the GNU General Public License.
1
www.satheesh.bigbig.com/EnggPhy 2
1 Introduction
Theturnofthe20
th
centurybroughtthestartofarevolutioninphysics. In1900,MaxPlanckpublished
his explanation of blackbody radiation. This equation assumed that radiators are quantized, which
proved to be the opening argument in the edi?ce that would become quantum mechanics. In this
chapter, many of the developments which form the foundation of modern physics are discussed.
2 Blackbody radiation spectrum
A blackbody is an object that absorbs all light that falls on it. Since no light is re?ected or transmitted,
the object appears black when it is cold. The term blackbody was introduced by Gustav Kirchho? in
1860. A perfect blackbody, in thermal equilibrium, will emit exactly as much as it absorbs at every
wavelength. The light emitted by a blackbody is called blackbody radiation.
Theplotofdistributionofemittedenergyasafunctionofwavelengthandtemperatureofblackbody
is know as blackbody spectrum. It has the following characteristics.
• The spectral distribution of energy in the radiation depends only on the temperature of the
blackbody.
• The higher the temperature, the greater the amount of total radiation energy emitted and also
energy emitted at individual wavelengths.
• The higher the temperature, the lower the wavelength at which maximum emission occurs.
Many theories were proposed to explain the nature of blackbody radiation based on classical physics
arguments. But non of them could explain the complete blackbody spectrum satisfactorily. These
theories are discussed in brief below.
www.satheesh.bigbig.com/EnggPhy 3
2.1 Stefan-Boltzmann law
The Stefan-Boltzmann law, also known as Stefan’s law, states that the total energy radiated per unit
surface area of a blackbody in unit time (known variously as the blackbody irradiance, energy ?ux
density, radiant ?ux, or the emissive power), E
?
, is directly proportional to the fourth power of the
blackbody’s thermodynamic temperature T (also called absolute temperature):
E
?
= sT
4
. (1)
The constant of proportionality s is called the Stefan-Boltzmann constant or Stefan’s constant. It is
not a fundamental constant, in the sense that it can be derived from other known constants of nature.
The value of the constant is 5.6704×10
-8
Js
-1
m
-2
K
-4
. The Stefan-Boltzmann law is an example of
a power law.
2.2 Wien’s displacement law
Wien’s displacement law states that there is an inverse relationship between the wavelength of the peak
of the emission of a blackbody and its absolute temperature.
?
max
?
1
T
T?
max
= b (2)
where
?
max
is the peak wavelength in meters,
T is the temperature of the blackbody in kelvins (K), and
bisaconstantofproportionality, calledWien’sdisplacementconstantandequals2.8978×10
-3
mK.
In other words, Wien’s displacement law states that the hotter an object is, the shorter the wavelength
at which it will emit most of its radiation.
2.3 Wien’s distribution law
According to Wein, the energy density, E
?
, emitted by a blackbody in a wavelength interval ? and
?+d? is given by
E
?
d? =
c
1
?
5
e
(-c
2
/?T)
d? (3)
where c
1
and c
2
are constants. This is known as Wien’s distribution law or simply Wein’s law. This
law holds good for smaller values of ? but does not match the experimental results for larger values of
?. Wien received the 1911 Nobel Prize for his work on heat radiation.
2.4 Rayleigh-Jeans’ law
According to Rayleigh and Jeans the energy density, E
?
, emitted by a blackbody in a wavelength
interval ? and ?+d? is given by
E
?
d? =
8pkT
?
4
d?, (4)
where k is the Boltzmann’s constant whose value is equal to 1.381×10
-23
JK
-1
.
It agrees well with experimental measurements for long wavelengths. However it predicts an en-
ergy output that diverges towards in?nity as wavelengths grow smaller. This was not supported by
experiments and the failure has become known as the ultraviolet catastrophe or Rayleigh-Jeans
catastrophe. Here the word ultraviolet signi?es shorter wavelength or higher frequencies and not the
ultraviolet region of the spectrum. One more thing to note is that, it was not, as is sometimes asserted
in physics textbooks, a motivation for quantum theory.
www.satheesh.bigbig.com/EnggPhy 4
2.5 Planck’s law of black-body radiation
Explaining the blackbody radiation curve was a major challenge in theoretical physics during the late
nineteenth century. All the theories based on classical ideas failed in one or the other way. The
wavelength at which the radiation is strongest is given by Wien’s displacement law, and the overall
power emitted per unit area is given by the Stefan-Boltzmann law. Wein’s law could explain the
blackbody radiation curve only for shorter wavelengths whereas Rayleigh-Jeans’ law worked well only
for larger wavelengths. The problem was ?nally solved in 1901 by Max Planck.
Planck came up with the following formula for the spectral energy density of blackbody radiation
in a wavelength range ? and ?+d?,
E
?
d? =
8phc
?
5
1
e
hc/?kT
-1
d?, (5)
where h is the Planck’s constant whose value is 6.626×10
-34
Js. This formula could explain the entire
blackbody spectrum and does not su?er from an ultraviolet catastrophe unlike the previous ones. But
the problem was to justify it in terms of physical principles. Planck proposed a radically new idea that
the oscillators in the blackbody do not have continuous distribution of energies but only in discrete
amounts. An oscillator emits radiation of frequency ? when it drops from one energy state to the next
lower one, and it jumps to the next higher state when it absorbs radiation of frequency ?. Each such
discrete bundle of energy h? is called quantum. Hence, the energy of an oscillator can be written as
E
n
= nh? n = 0,1,2,3,.... (6)
2.5.1 Derivation Wien’s law from Planck’s law
The Planck’s law of blackbody radiation expressed in terms of wavelength is given by
E
?
d? =
8phc
?
5
1
e
hc/?kT
-1
d?.
In the limit of shorter wavelengths, hc/?kT becomes very small resulting in
e
hc/?kT
 1.
Therefore
e
hc/?kT
-1˜ e
hc/?kT
.
This reduces the Planck’s law to
E
?
d? =
8phc
?
5
1
e
hc/?kT
d?
or
E
?
d? =
8phc
?
5
e
-hc/?kT
d?.
Now identifying 8phc as c
1
and hc/k as c
2
, the above equation takes the form
E
?
d? =
c
1
?
5
e
(-c
2
/?T)
d?.
This is the familiar Wien’s law.
www.satheesh.bigbig.com/EnggPhy 5
2.5.2 Derivation of Rayleigh-Jeans’ law from Planck’s law
The Planck’s law of blackbody radiation expressed in terms of wavelength is given by
E
?
d? =
8phc
?
5
1
e
hc/?kT
-1
d?.
In the limit of long wavelengths, the term in the exponential becomes small. Now expressing it in
the form of power series (e
x
= 1+x+
x
2
2!
+
x
3
3!
+
x
4
4!
+....),
e
hc/?kT
= 1+

hc
?kT

+

hc
?kT

2
2!
+...
Since
hc
?kT
is small, any higher order of the same will be much smaller, so we truncate the series beyond
the ?rst order term,
e
hc/?kT
˜ 1+

hc
?kT

.
Therefore, the Planck’s law takes the form
E
?
d? =
8phc
?
5
1
(1+hc/?kT)-1
d?,
that is,
E
?
d? =
8phc
?
5
1
(hc/?kT)
d?,
E
?
d? =
8phc
?
5
?kT
hc
d?.
This gives back the Rayleigh-Jeans Law
E
?
d? =
8pkT
?
4
d?.
3 Photo-electric e?ect
The phenomenon of electrons being emitted from a metal when struck by incident electromagnetic
radiation of certain frequency is called photoelectric e?ect. The emitted electrons can be referred to as
photoelectrons. The e?ect is also termed the Hertz E?ect in the honor of its discoverer, although the
term has generally fallen out of use.
3.1 Experimental results of the photoelectric emission
1. The time lag between the incidence of radiation and the emission of a photoelectron is very small,
less than 10
-9
second.
2. For a given metal, there exists a certain minimum frequency of incident radiation below which
no photoelectrons can be emitted. This frequency is called the threshold frequency or critical
frequency, denoted by ?
0
. The energy corresponding to this threshold frequency is the mini-
mum energy required to eject a photoelectron from the surface. This minimum energy is the
characteristic of the material which is called work function (f).
3. For a given metal and frequency of incident radiation, the number of photoelectrons ejected is
directly proportional to the intensity of the incident light.
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