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 â€¢ Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” 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 â€¢ Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” 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. â€¢ The spectral distribution of energy in the radiation depends only on the temperature of the blackbody. â€¢ The higher the temperature, the greater the amount of total radiation energy emitted and also energy emitted at individual wavelengths. â€¢ 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 â€¢ Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” 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. â€¢ The spectral distribution of energy in the radiation depends only on the temperature of the blackbody. â€¢ The higher the temperature, the greater the amount of total radiation energy emitted and also energy emitted at individual wavelengths. â€¢ 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 Stefanâ€™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 blackbodyâ€™s thermodynamic temperature T (also called absolute temperature): E ? = sT 4 . (1) The constant of proportionality s is called the Stefan-Boltzmann constant or Stefanâ€™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 Wienâ€™s displacement law Wienâ€™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, calledWienâ€™sdisplacementconstantandequals2.8978×10 -3 mK. In other words, Wienâ€™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 Wienâ€™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 Wienâ€™s distribution law or simply Weinâ€™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-Jeansâ€™ 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 Boltzmannâ€™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 â€¢ Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” 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. â€¢ The spectral distribution of energy in the radiation depends only on the temperature of the blackbody. â€¢ The higher the temperature, the greater the amount of total radiation energy emitted and also energy emitted at individual wavelengths. â€¢ 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 Stefanâ€™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 blackbodyâ€™s thermodynamic temperature T (also called absolute temperature): E ? = sT 4 . (1) The constant of proportionality s is called the Stefan-Boltzmann constant or Stefanâ€™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 Wienâ€™s displacement law Wienâ€™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, calledWienâ€™sdisplacementconstantandequals2.8978×10 -3 mK. In other words, Wienâ€™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 Wienâ€™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 Wienâ€™s distribution law or simply Weinâ€™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-Jeansâ€™ 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 Boltzmannâ€™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 Planckâ€™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 Wienâ€™s displacement law, and the overall power emitted per unit area is given by the Stefan-Boltzmann law. Weinâ€™s law could explain the blackbody radiation curve only for shorter wavelengths whereas Rayleigh-Jeansâ€™ 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 Planckâ€™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 Wienâ€™s law from Planckâ€™s law The Planckâ€™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 -1Ëœ e hc/?kT . This reduces the Planckâ€™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 Wienâ€™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 â€¢ Arthur Beiser, Concepts of Modern Physics, 6 th Edition, Tata McGraw-Hill Pub- lishing Company Limited, ISBN- 0-07-049553-X. â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€”â€” 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. â€¢ The spectral distribution of energy in the radiation depends only on the temperature of the blackbody. â€¢ The higher the temperature, the greater the amount of total radiation energy emitted and also energy emitted at individual wavelengths. â€¢ 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 Stefanâ€™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 blackbodyâ€™s thermodynamic temperature T (also called absolute temperature): E ? = sT 4 . (1) The constant of proportionality s is called the Stefan-Boltzmann constant or Stefanâ€™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 Wienâ€™s displacement law Wienâ€™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, calledWienâ€™sdisplacementconstantandequals2.8978×10 -3 mK. In other words, Wienâ€™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 Wienâ€™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 Wienâ€™s distribution law or simply Weinâ€™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-Jeansâ€™ 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 Boltzmannâ€™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 Planckâ€™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 Wienâ€™s displacement law, and the overall power emitted per unit area is given by the Stefan-Boltzmann law. Weinâ€™s law could explain the blackbody radiation curve only for shorter wavelengths whereas Rayleigh-Jeansâ€™ 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 Planckâ€™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 Wienâ€™s law from Planckâ€™s law The Planckâ€™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 -1Ëœ e hc/?kT . This reduces the Planckâ€™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 Wienâ€™s law. www.satheesh.bigbig.com/EnggPhy 5 2.5.2 Derivation of Rayleigh-Jeansâ€™ law from Planckâ€™s law The Planckâ€™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 Ëœ 1+ hc ?kT . Therefore, the Planckâ€™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.Read More

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