Dual Nature of Radiation and Matter

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Questions:

Generic 5

11 x 5 = 55

The main aim of Davisson and Germer was to study about nickel surface by directing beam of electrons at its surface and note the number of electrons that bounced off at different angles. The carried out their experiment inside a vacuum chamber where an air after entering the chamber gives an oxide film on nickel surface. Again the does their experiment and found that the electrons which hits the nickel surface were scattered by atoms that appears from crystal planes in nickel crystal. By doing their regular experiment, Davisson and Germer's at once found diffraction of electrons which was an initial proof that confirm about de Broglie's hypothesis and shows wave properties in particles.
(a) What can we infer about the values shown by Davisson and Germer?
(b) Write the expression to find the wavelength of an electron when accelerated through a potential difference of V volts.

Shivam had knowledge that energy and momentum of an electron are related to frequency and wavelength of an electron are related to frequency and wavelength of the associated matter (de-Broglie) wave by the relations, E = hv, \(p=\frac { h }{ \lambda } \). But when he heard from his friend that value of \(\lambda \) is physically significance but the value of frequency v has no physical significance. Shivam requested Shubham to explain such thing clearly. Shubham said to him that according to de-Broglie a particle behavies as a wave, but it is now established that a particle cannot be equivalent to a single wave, but it is equivalent to a group of waves or a wave packet.
As \(p=\frac { h }{ \lambda } \) so in the discussion of matter waves, a wave packet is significant and hence only wavelength, \(\lambda \) is significant. As a single wave is insignificant, so phase velocity (velocity of a single wave, v = v\(\lambda \)) is insignificant and thus, frequency v is also insignicant.
(a) What are the values shown by Shubham?
(b) Calculate the
(i) momentum and
(ii) de Broglie wavelength of the electrons accelerated throught a potential difference of 56 V.

Davisson and Germer's actual objective was to study the surface of a piece of nickel by directing a beam of electrons at the surface and observing how many electrons bounced off at various angles. Though the ecperiment was conducted in a vacuum chamber, air entered the chamber, producing an oxide flim on the nickel surfec. When they started the experiment again and the electrons hit the surface, they were scattered by atoms which originated from crystal planes inside the nickel crystal. Davisson and Germer's accidental discovery of the diffraction of electrons was the first direct evidence confirming de Broglie's hypothesis that particles can have wave properties as well.
(a) What can we infer about the values shown by Davisson and Germer?
(b) Write the expression to find the wavelength of an electron when accelerated through a potential difference of V volts.

Ram knows that red light has greater and so it is much bright, but in case of photoelectric emission it cannot produce the emission of electrons from a clean zinc surface, while even weak ultraviolet radiation can do so. He could not know specific cause of such thing. Then he went to his friend Shyam for its specific explanation. Shyam explained him that the photoemission of electron does not depend on the intensity while it depends on the frequency and thus on the energy of photon of incident light. The energy of photon of red light cannot emit photoelectrons. Similarly, the energy of photon of ultraviolet light is greater than the work function of zinc, so ultraviolet light can emit photoelectrons.
(a) What values are noticed in Shyam?
(b) The work functions of lithium and copper are 2.3eV and 4eV respectively. Which of these metals are useful for the photoelectric cell working with visible light? Explain.

Mohan thought that there are materials which absorb photons of shorter wavelength and emit photons of longer wavelength. But, can there be stable substances which absorb photons of larger wavelength and emit light of shorter wavelength? He got confused and could not find its answer. Then he requested his friend Sohan. Sohan explained him that in the first case, energy given out is less than the energy supplied. But in the second case, the material has to supply the energy as the emitted photon has to supply the energy as the emitted photon has more energy, which cannot happen for stable substances.
(a) What values do you notice in Sohan?
(b) Consider a metal exposed to light of wavelength 600 nm. The maximum energy of the electron doubles when light of wavelength 400 nm is used. Find the work function in eV.

Neha's brother was riding the bike on the highway and she was sitting behind him. While sitting, at a place traffic signal turned red from green and her brother continued riding without noticing the signal change. Neha observed the whole situation and asked her brother to stop. Her brother felt happy on his sister's intelligence.
Read above passage and answer the following questions:
(i) Why Neha's brother became happy? What kind of value is expressed by Neha?
(ii) What are the principles that are used in maintaining traffic signals?
(iii) What is the leading physical quantity i8n the process? Write an equation for the speed of the photoelectron.

"Know your face beauty through comp[lexion meter" was one of the stall on science exhibition. A student interested to know his/her face beauty was made to stand on a platform and light from a lamp was made to fall on his/her face. The reading of complexion meter indicated the face beauty of the student which might be very fair, fair,semi-fair, semidark dark etc.
(i) What is the basic concept used in the working of complexion meter?
(ii) How is the face beauty recorded by face complexation meter?
(iii) What basic values do you learn from the above study?

Radiation has dual nature,i.e., it possesses the properties of both; wave and particle.This prompted de-Broglie to predict dual nature of moving material particles.Thus waves are associated with moving material particles which are called matter waves. The wavelength of matter wave is given by \(\lambda =\frac { h }{ mv } \), where m is the mass, v is the speed of the particle and h is Plank's constant. Read the above paragraph and answer the following questions;
(i) How was the wave nature of the electron established?
(ii) What are the de-Broglie wavelength associated with a particle (i) at rest (ii) moving with infinite speed?
(iii) What are the basic values displayed with this study?

By how much would the stopping potential for a given photosensitive surface go up if the frequency of the incident radiations were to be increased from \(6\times { 10 }^{ 15 }Hz \ to \ 16\times { 10 }^{ 15 }Hz\) ?Given \(h=6.4\times { 10 }^{ -34 }Js,e=1.6\times { 10 }^{ -19 }C \ and \ c=3\times { 10 }^{ 8 }m/s\)

10.

Work function of sodium is 2.3eV. Does sodium photoelectric emission for orange light \((\lambda =6800\overset { \circ }{ A } )\)? Given \(h=6.63\times { 10 }^{ -34 }Js\)

11.

Consider a thin target \(({ 10 }^{ -2 }\)m square, \({ 10 }^{ -3 }\)m thickness) of sodium, which produces a photocurrent of \(100\mu A\) when a light of intensity 100w/\({ m }^{ 2 }\) ( \((\lambda =660nm)\) falls on it. Find the probability that a photoelectron is produced when a photon strikes a sodium atom.[Take density of Na = 0.97 kg/\({ m }^{ 3 }\)

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Answers Key:

Generic 5

11 x 5 = 55

(a) Perseverance, not giving up and patience.
(b) \(\lambda =\frac { 12.27 }{ \sqrt { V } } \)\(\overset { o }{ A } \)

(a) The values shown by Shubham are:
(i) High degree of general awarness.
(ii) High order of thinking skills.
(iii) Concern for his friend.
(iv) Helping and caring nature.
(b) Mass of electron, \(m=9.1\times { 10 }^{ -31 }kg\)
(i) Momentum, \(p=\sqrt { { 2mE }_{ k } } =\sqrt { 2meV } \)
\(=\sqrt { 2\times 9.1\times { 10 }^{ -31 }\times 1.6\times { 10 }^{ -19 }\times 56 }\)
\( =4.04\times { 10 }^{ -24 }kg{ ms }^{ -1 }\)
(ii) de Broglie wavelength,
\(=\frac { h }{ p } =\frac { 6.63\times { 10 }^{ -34 } }{ 4.04\times { 10 }^{ -24 } } \)
\( =1.64\times { 10 }^{ -10 }m=0.164nm\)

(a) Perseverance, not giving up and patience.
(b) \(\lambda =\frac { 12.27 }{ \sqrt { V } } \overset { 0 }{ A } \)

(a) The values noticed in Shyam are:
(i) High degree of general awareness.
(ii) Concern for his friend.
(iii) Helping and caring nature.
(b) The threshold wavelength, \({ \lambda }_{ 0 }=\frac { hc }{ W } \)
For lithium, \({ \lambda }_{ 0 }=\frac { 12375 }{ 2.3 } \overset { 0 }{ A } =5380\overset { 0 }{ A } \)
For copper, \({ \lambda }_{ 0 }=\frac { 12375 }{ 4 } \overset { 0 }{ A } =3094\overset { 0 }{ A } \)
The wavelength 5380\(\overset { 0 }{ A } \) lies in visible region, thus lithium will be useful for photoelectric cell.

(a) The value noticed in Sohan are:
(i) High degree of general awareness.
(ii) Concern for his friend.
(iii) Helping and caring nature.
(b) Maximum energy \(=hv-\phi o\)
\(\Rightarrow \quad \left( \frac { 1230 }{ 600 } -\phi o \right) =\frac { 1 }{ 2 } \left( \frac { 1230 }{ 400 } -\phi o \right) \)
\(\Rightarrow \frac { 1230 }{ 300 } -2\phi o=\frac { 1230 }{ 400 } -\phi o=\frac { 1230 }{ 1200 } \)
= 1.02 eV

(i) Neha's brother was proud and happy on his sister's presence of mind, awareness of traffic rules with high visual sensibility.
(ii) The principle of photoelectric effect is used to maintain the traffic signal.
(iii) The leading physical quantity of photoelectric current is arisen due to the motion of photoelectrons.
The required equation is \(v=\sqrt { \frac { 2E }{ m } -\frac { 2\phi }{ m } } \)
where, v = speed of the electron, E = energy of the photon, m = mass of the electron and \(\phi \) = work function for the metal.

(i) The basic concept used in the working of complexation meter is a photoelectric effect.
(ii) The reading of complexation meter depends on the current generated from the light reflected from the face. Fairer the colour more is the light reflected and vice versa.
(iii) Complexation meters judge your beauty only. The inner beauty of a person is judged by the virtues he or she posseses. In my opinion, inner beauty is much more valuable than the face beauty.

(i) Davisson and Germer observed diffraction patterns of slow moving electrons. And G.P. Thomson observed a diffraction pattern of fast moving electrons. As diffraction is essentially a wave phenomenon, therefore, it was concluded that wave must be associated with moving electrons.
(ii) (a) At rest, v = 0, \(=\frac { h }{ mv } =\frac { h }{ m\times 0 } =\infty \)
(b) Particle moving with infinite speed, v = \(\infty \)
\(\lambda =\frac { h }{ mv } =\frac { h }{ m\times \infty } \)
(iii) The dual nature of moving material particles reveals in a way the nature of Almighty God. He is in a visible form (i.e., Sakar) like a visible particle and also without any form (i.e., Nirakar) like a wave.It depends on us how we realize him.

\(Here, \ { v }_{ 1 }=6\times { 10 }^{ 15 }Hz,{ v }_{ 2 }=16\times { 10 }^{ 15 }Hz,{ V }_{ 02 }-{ V }_{ 01 }=?\)
\( As \ \ e{ V }_{ 0 }=hv-{ \phi }_{ 0 }\)
\(\therefore \ { eV }_{ 01 }=h{ v }_{ 1 }-{ \phi }_{ 0 } \ and \ { eV }_{ 01 }=h{ v }_{ 2 }-{ \phi }_{ 0 }\)
\(\therefore \ e\left( { V }_{ 02 }-{ V }_{ 01 } \right) =h\left( { v }_{ 2 }-{ v }_{ 1 } \right) \)
\( or \ { V }_{ 02 }-{ V }_{ 01 }=\frac { h }{ e } { v }_{ 2 }-{ v }_{ 1 }\)
\( { V }_{ 02 }-{ V }_{ 01 }=\frac { \left( 6.4\times { 10 }^{ -34 } \right) }{ 1.6\times { 10 }^{ -19 } } \times \left( 16\times { 10 }^{ 15 }-6\times { 10 }^{ 15 } \right) =40V\)

10.

\(Here,{ \phi }_{ 0 }=2.3eV\)
Wavelength of orange light,
\((\lambda =6800\overset { \circ }{ A } )=6800\times { 10 }^{ -10 }m\)
The energy of the photon of orange light is given by,
\(E=\frac { hc }{ \lambda } =\frac { 6.63\times { 10 }^{ -34 }\times 3\times { 10 }^{ 8 } }{ 6800\times { 10 }^{ -10 } } joules\)
\(=\frac { 6.63\times 3 }{ 68 } \times \frac { { 10 }^{ -18 } }{ 1.6\times { 10 }^{ -19 } } eV\)
\( =1.83eV\)
Which is less than\({ \phi }_{ 0 }\). Since the energy of the incident photon of orange light is less than the work function for sodium, therefore, photoelectric emission from sodium can not take place with orange.

11.

6 x \({ 10 }^{ 26 }\) Na atoms weights 23 kg.
Volume of target = \({ (10 }^{ -4 }\times{ 10 }^{ -3 })={ 10 }^{ -7 }\) \({ m }^{ 3 }\)
Density of sodium = (d) = 0.97 kg/\({ m }^{ 3 }\)
Volume of \({ 6\times10 }^{ 26 }\) Na atoms = \(\frac { 23 }{ 0.97 } { m }^{ 3 }=23.7{ m }^{ 3 }\)
Volume of occupied of 1 Na atom = \(\frac { 23 }{ 0.97\times6\times{ 10 }^{ 26 } } { m }^{ 3 }\)
\(=3.95\times{ 10 }^{ -26 }=2.53\times{ 10 }^{ 18 }\)
No of sodium atoms in the target
\(=\frac { { 10 }^{ -7 } }{ 3.95\times{ 10 }^{ -26 } } =2.53\times{ 10 }^{ 18 }\)
Number of photons/in the beam for \({ 10 }^{ -4 }{ m }^{ 2 }=n\) Energy per second
\(=nhv={ 10 }^{ -4 }J\times100={ 10 }^{ -2 }W\)
\(hv(for\lambda =660nm)=\frac { 1234.5 }{ 600 }\)
\(=2.05eV=2.05\times1.6\times{ 10 }^{ -19 }\)
\(\\ =3.28\times{ 10 }^{ -19 }J\)
\(n\frac { { 10 }^{ -2 } }{ 3.28\times{ 10 }^{ -19 } } =3.05\times{ 10 }^{ 16/s }\)
\( n=\frac { 1 }{ 3.2 } \times{ 10 }^{ 17 }=3.1\times{ 10 }^{ 16 }\)
If p is the probability of emission per atom , per photon,the number of photoelectrons emitted/second
\(=p\times3.1\times{ 10 }^{ 16 }\times2.53\times{ 10 }^{ 18 }\)
Current \(= p\times3.1\times{ 10 }^{ +16 }\times2.53\times{ 10 }^{ 18 }\times1.6\times{ 10 }^{ -19 }A\)
This must equal \(100\mu A\)
\(p=\frac { 100\times{ 10 }^{ -6 } }{ 1.25\times{ 10 }^{ +16 } } \)
\(\therefore \) \(p=8\times{ 10 }^{ -21 }\)