SCHEME OF EVAUATION MANIPA INSTITUTE OF TECHNOOGY MANIPA UNIVERSITY, MANIPA SECOND SEMESTER B.Tech. END-SEMESTER EXAMINATION - MAY SUBJECT: ENGINEERING PHYSICS (PHY/) Time: 3 Hrs. Max. Marks: 5 Note: Answer any FIVE FU questions. Each question carries marks Answer all the sub questions of a main question in a continuous sequence. Write specific and precise answers. Any missing data may suitably be assumed. Write question number on the margin only. Draw neat sketches wherever necessary. Physical Constants: Speed of light in vacuum = 3. X 8 m/s Electron charge =.6 X 9 C Electron mass = 9. X 3 kg Atomic mass unit (u) =.66 X 7 kg Boltzmann constant =.38 X 3 J/ K Planck s constant = 6.63 X 3 J.s A. Obtain an expression for the radius of m TH order bright ring in the case of Newton s rings. [5] B. Give a brief account of quantum model of H-atom. [3] C. Sketch schematically the plot of potential energy and its components as a function of internuclear separation distance for a system of two atoms. [] A. Explain the experiment on compton effect. [5] B. Explain the origin of characteristic X-ray spectrum with a sketch of X-ray energy level diagram. [3] C. Explain Rayleigh s criterion for resolving images due to a circular aperture. [] 3A. Explain briefly the energy band theory of solids. [5] 3B. Write the expression for intensity of light at an angular position θ on the screen due to the diffraction of light at a single-slit. Show that the intensity at the first secondary maximum is.5% of the intensity at the central maximum. [3] 3C. Which are the mathematical features of a wave function? [] /
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION A. A diffraction grating (. mm wide) has rulings/mm and a principal maximum is noted at θ = 8.. (i) What are the possible wavelengths of the incident visible light (-7nm)? (ii) What are the values of the dispersion of the grating in the vicinity of these lines? [5] E3-8 E3-3 B. ithium has work function of.3 ev. ight with a wavelength of nm is incident on this metal. Determine the stopping potential. [3] P-5b C. Each atom of gold contributes one free-electron to the metal. The free electron density in gold is 5.9 x 8 /m 3. The free electron density (n e) in a metal is given E3- by the expression:. Compute the Fermi energy [E F()] of gold. [] 5A. A particle, in ground-state of a one-dimensional box has a wave function (inside the P-8 MODIFIED box) given by ( = length of the box) : Calculate the probability of finding the particle between x = and x = /. [5] 5B. A three level laser of the type shown in figure, emits laser light at a wavelength of 55 nm, near the centre of the visible band. At what temperature the ratio of the population of the atoms in the upper level E to that in the lower level E becomes half? [3] SP8-7b PUMPING TRANSITION RAPID DECAY METASTABE E 3 ASER TRANSITION E E 5C. A sheet of glass having an index of refraction of. is to be coated with a film of material having an index of refraction.55 such that green light (λ = 55 nm) is preferentially transmitted. What is the minimum thickness of the film that will achieve the result? [] P-8 6A. Calculate the moment of inertia of an NaCl molecule about its centre of mass. The atoms are separated by a distance of 8 pm. Calculate the wavelength of the radiation emitted when an NaCl molecule undergoes a rotational transition from the J = state to the J = state. Atomic mass of sodium is 3. u and atomic mass of chlorine is 35.5 u. [5] P3-6B. Unpolarized light falls on two ideal polarizing sheets placed one on top of the other. What must be the angle between the transmission axes of the two sheets if the intensity of the transmitted light is one-third the intensity of the incident beam? [3] E- 6C. Find the wavelength of the photon resulting from the transition of a Rydberg atom from n = 73 level to n = 7. What is the radius of the electron orbit for a Rydberg atom for which n = 73? Rydberg constant =.97 x 7 /m, Bohr radius = 5.9 pm [] E-b,c - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION A. Obtain an expression for the radius of m TH order bright ring in the case of Newton s rings. [5] Constructive interference d = (m - ½), (n = for air film) d R R d R R r R r r R R R r R using binomial expansion... r R Substituting d in d = (m - ½) we get r (m ) R m,,... (maxima) B. Give a brief account of quantum model of H-atom. [3] kee The potential energy function for the H-atom is U(r) r k e is Coulomb constant, r is radial distance of electron from proton [H-nucleus]. The time-independent schrodinger equation in 3-dimensional space is U E m x y z Since U has spherical symmetry, it is easier to solve the schrodinger equation in spherical polar coordinates (r, θ, φ): z where r x y z P θ is the angle between z-axis and r, θ r φ is the angle between the x-axis and the projection of onto the xy-plane. y r φ It is possible to separate the variables r, θ, φ as follows: x (r, θ, φ) = R(r) f(θ) g(φ) By solving the three separate ordinary differential equations for R(r), f(θ), g(φ), with conditions that the normalized and its first derivative are continuous and finite everywhere, one gets three different quantum numbers for each allowed state of the H-atom. The quantum numbers are integers and correspond to the three independent degrees of freedom. 3
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION C. Sketch schematically the plot of potential energy and its components as a function of internuclear separation distance for a system of two atoms. [] for proper labelling of axes for proper shape and labelling of repulsive potential and attractive potential for proper shape and labelling of total potential for proper labelling of Binding energy and Equilibrium separation distance
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION A. Explain the experiment on compton effect. [5] Here X- ray photons are scattered through 9 from a carbon target. The wavelength is measured with a rotating crystal spectrometer using Bragg s law. Intensity of the scattered X-rays are measured using the ionization chamber. Contrary to the classical predictions where X-rays are treated as waves, in Compton experiment, at a given angle, only one frequency for scattered radiation is seen. This is shown in the figure, scattered x-ray intensity versus wavelength for Compton scattering at =, 5, 9, and 35. Compton could explain the experimental result by taking a billiard ball type collisions between particles of light (X-ray photons) and electrons in the material. The graphs for three nonzero angles show two peaks, one at o and one at > o. The shifted peak at is caused by the scattering of X-rays from free electrons. Shift in wavelength was predicted by Compton to depend on scattering angle as ' o m h c ( cos) This is known as Compton shift equation, and the factor is m h c called the Compton wavelength. Prediction were in excellent agreement with the experimental results. 5
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION B. Explain the origin of characteristic X-ray spectrum with a sketch of X-ray energy level diagram. [3] The peaks in the x-ray spectrum have wavelengths characteristic of the target element in the x-ray tube and hence they form the characteristic x-ray spectrum. When a high energy (K = e V, V = x-ray tube voltage) electron strikes a target atom and knocks out one of its electrons from the inner shells with energy E nf ( E nf K, n f = integer), the vacancy in the inner shell is filled up by an electron from the outer shell (energy = E ni, n i = integer). The characteristic x-ray photon emitted has the energy: hc hf E ni E nf A K x-ray results due to the transition of the electron from -shell to K-shell. A K x-ray results due to the transition of the electron from M-shell to K-shell. When the vacancy arises in the -shell, an -series (,, ) of x-rays results. Similarly, the origin of M-series of x-rays can be explained. C. Explain Rayleigh s criterion for resolving images due to a circular aperture. [] Raleigh s criterion for optical resolution: The images of two closely spaced sources is said to be just resolved if the angular separation of the two point sources is such that the central maximum of the diffraction pattern of one source falls on the first minimum of the diffraction pattern of the other. 6
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION 3A. Explain briefly the energy band theory of solids. [5] for each main point with a max of 5 marks There are two different wave functions (r) and (r) S S (for an atom with single s-electron outside of a closed shell) for which the probability density is the same. The wave functions of two atoms combine to form a composite wave function for the two-atom system when the atoms are close together. In Figure (a), two atoms with wave functions (r) combine. In S Figure (b), two atoms with wave functions (r) and (r) S S combine. These two possible combinations of wave functions represent two possible states of the two-atom system which have slightly different energies. Thus, each energy level of an atom splits into two close energy levels when the wave functions of the two atoms overlap. ENERGY BANDS OF SODIUM CRYSTA 3p 3s p E Figure (a) shows the splitting of s and s levels when two sodium atoms are brought together. Figure (b) shows the splitting of s and s levels into five levels each when five sodium atoms are brought together. Figure (c) shows formation of energy bands (bunches of closely spaced energy levels) when a large number of sodium atoms are assembled to form a solid. Thus, in a crystalline solid there are a large number of allowed energy bands that arise from the various atomic energy levels. Forbidden energy gaps occur between the allowed bands. Some bands may be wide enough in energy so that there is an overlap between the adjacent bands. The s, s, and p bands of solid sodium are filled completely with electrons. Since there are (+) energy states in a subshell each energy band has (+)N energy states, where N = number of atoms in the crystal. The 3s band (N states) of solid sodium has only N electrons and is partially full; The 3p band and the bands above this are completely empty. s s 7
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION 3B. Write the expression for intensity of light at an angular position θ on the screen due to the diffraction of light at a single-slit. Show that the intensity at the first secondary maximum is.5% of the intensity at the central maximum. [3] Intensity of light at an angular position θ on the screen due to the diffraction of light at a single-slit: I m = central maximum intensity, a = slit width, λ = wavelength of light For secondary maxima, α = (m+) π /, m =,, 3,..., approximately. At first secondary maximum, m =, α = 3 π /, sin α =., ie, I θ = (.5%) I m 3C. Which are the mathematical features of a wave function? [] (i) (ii) (iii) (iv) x The important mathematical features of a physically reasonable wave function (x) for a system are (x) may be a complex function or a real function, depending on the system; (x), must be finite, continuous and single valued everywhere; The space derivatives of, must be finite, continuous and single valued everywhere; must be normalizable. 8
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION A. A diffraction grating (. mm wide) has rulings/mm and a principal maximum is noted at θ = 8.. (i) What are the possible wavelengths of the incident visible light (-7nm)? (ii) What are the values of the dispersion of the grating in the vicinity of these lines? [5] E3-8 E3-3 θ = 8. M = rulings/mm d = /M = 5 nm d sin θ = m λ λ = (d sin θ) / m = (37 nm) / m m = => λ = 7 nm m = 3 => λ = 78 nm m = => λ = 587 nm m = 5 => λ = 69 nm m = 6 => λ = 39 nm D = m / (d cos θ) D = /( d cos 8. ) = D 5 = 5 /( d cos 8. ) = 9.6 x 5 rad/m = 9.6 x rad/nm =.59 /nm.3 x 6 rad/m =.3 x 3 rad/nm =.69 /nm B. ithium has work function of.3 ev. ight with a wavelength of nm is incident on this metal. Determine the stopping potential. [3] P-5b Φ =.3 ev λ = nm e =.6 X 9 C h = 6.63 X 3 J.s = 3.68 X 9 J c = 3. X 8 m/s K MAX = h f φ e (ΔV S ) = (h c / λ) φ ΔV S = [(h c / λ) φ] / e = [.9 X 9 J] / e =.86 V mark C. Each atom of gold contributes one free-electron to the metal. The free electron density in gold is 5.9 x 8 /m 3. The free electron density (n e) in a metal is given E3- by the expression:. Compute the Fermi energy [E F()] of gold. [] m = 9. X 3 kg n e = 5.9 x 8 /m 3 h = 6.63 X 3 J.s E F 3 h 3 ne m 8 = 8.85 x 9 J = 5.53 ev 9
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION 5A. A particle, in ground-state of a one-dimensional box has a wave function (inside the P-8 MODIFIED box) given by ( = length of the box) : Calculate the probability of finding the particle between x = and x = /. [5] Pr obability x dx Probability =.9 x cos dx ] x sin x sin sin ] dx dx x cos dx 5B. A three level laser of the type shown in figure, emits laser light at a wavelength of 55 nm, near the centre of the visible band. At what temperature the ratio of the population of the atoms in the upper level E to that in the lower level E becomes half? [3] SP8-7b PUMPING TRANSITION RAPID DECAY METASTABE E 3 ASER TRANSITION E E λ = 55 nm k =.38 X 3 J/K h = 6.63 X 3 J.s c = 3. X 8 m/s n(e) = exp ( E/kT) E E = h c / λ = 3.66 x 9 J, = 38 K 5C. A sheet of glass having an index of refraction of. is to be coated with a film of material having an index of refraction.55 such that green light (λ = 55 nm) is preferentially transmitted. What is the minimum thickness of the film that will achieve the result? [] P-8 n =.55 λ = 55 nm d = m λ n = (m λ)/n m = for min thickness d MIN = λ / ( n) = 69 nm
MU - MIT I BTECH SECOND SEMESTER- END EXAMINATION MAY ENGINEERING PHYSICS SCHEME OF EVAUATION 6A. Calculate the moment of inertia of an NaCl molecule about its centre of mass. The atoms are separated by a distance of 8 pm. Calculate the wavelength of the radiation emitted when an NaCl molecule undergoes a rotational transition from the J = state to the J = state. Atomic mass of sodium is 3. u and atomic mass of chlorine is 35.5 u. [5] P3- r = 8 pm m = 3. u m = 35.5 u u =.66 x 7 kg c = 3. X 8 m/s h = 6.63 X 3 J.s I = µ r =.8 x 5 kg.m [ ] =.6 m =.6 cm mark 6B. Unpolarized light falls on two ideal polarizing sheets placed one on top of the other. What must be the angle between the transmission axes of the two sheets if the intensity of the transmitted light is one-third the intensity of the incident beam? [3] E-,, =>, = 35.3 6C. Find the wavelength of the photon resulting from the transition of a Rydberg atom from n = 73 level to n = 7. What is the radius of the electron orbit for a Rydberg atom for which n = 73? Rydberg constant =.97 x 7 /m, Bohr radius = 5.9 pm [] P-b,c R H =.97 x 7 /m a o = 5.9 pm, n i = 73, n f = 7, λ =.9 m r n = n a o => r 73 = 73 a o = 3.9 µm - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -