Model Question Paper ENGINEERING PHYSICS (14PHY12/14PHY22) Note: Answer any FIVE full questions, choosing one full question from each module.

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1 Model Question Paper ENGINEERING PHYSICS (14PHY1/14PHY) Time: 3 hrs. Max. Marks: 100 Note: Answer any FIVE full questions, choosing one full question from each module. MODULE 1 1) a. Explain in brief Compton effect on the basis of quantum hypothesis.what is its physical significance? (6 Marks) b. Explain the terms Probability density, Normalization. Mention any two characteristics of wave function. (6 Marks) c. Obtain relationship between Group velocity and phase velocity. (4 Marks) OR ) a. Set up the one dimensional time independent Schrodinger wave equation. (7 Marks) b. State De Broglie hypothesis. Show that the De Broglie wavelength of an electron is found to be equal to 1.6 V A 0. (6 Marks) c. State and explain Heisenberg s Uncertainty principle. (3 Marks) d. The position and momentum of 1keV electron are simultaneously determined. If its position is located within 1A0, find the uncertainty in the determination of its momentum. (4 Marks) Soln. 1: a. Compton Effect is the decrease in energy or increase in wavelength of photon (an X-ray or gamma ray) when it interacts with matter. The amount by which the wavelength increases is called the Compton shift. When high energy photons interact with electron(s) at rest part of the energy is transferred to electron, and a photon containing the remaining energy being scattered in different direction so that the overall momentum of the system is conserved. The Compton scattering equation is given by, where λ λ = (1 cosθ) m 0 c

2 λ is the wavelength of the photon before scattering, λ is the wavelength of the photon after scattering, m 0 is the mass of the electron, θ is the angle by which the photons scattered, h is Planck s constant, and c is the speed of light. m 0 c = m is known as Compton Wavelengt. Physical Significance: 1. Compton Effect demonstrates that light cannot be explained purely as a wave phenomenon.. Light must behave as if it consists of particles in order to explain the Compton scattering. b. Probability Density: It measures the probability of the presence of the particle in the give region or volume or space. It is represented by ψ ψ or ψ. Normalization Condition: The total probability of finding the particle somewhere is unity i.e., ψ dxdydz = 1 The wave function ψ satisfying this condition is said to be normalized. Characteristics of Wave function: According to Max-Born, i. ψ doesn t measure the particle density at any point but gives the probability of finding the particle at that point at any instant of time ii. The value of square of wave function may be real or imaginary depending upon the value of ψ. Since the probability of finding a particle at given point in space must be real, it is taken as ψ ψ or ψ. iii. The probability of the particle being present in a volume dxdydz is ψ dxdydz. c. Relation between group velocity and phase velocity: The group velocity of a matter wave is given by whereas phase velocity is given by = dω dk v pase = ω k

3 From the definition of phase velocity, we can write ω = v pase k Substituting this in the expression for group velocity, we get = v pase k dk On differentiation, We rewrite this in the following form where = v pase + k dv pase dk k = π λ Substituting for k and dk/dλ we get, = v pase + k dv pase dλ and dk dλ = π λ dλ dk = v pase + π λ dv pase dλ λ π Further simplifying, = v pase λ dv pase dλ Soln.: a. Schrodinger s One Dimensional Time Independent Wave Equation: To derive the Schrodinger s time independent wave equation which describes the motion of electron in a potential V, one can approach the de-broglie s hypothesis. According to de-broglie s hypothesis the wavelength associated with an electron of charge e and mass m moving with velocity v in a potential V will be λ = mv = p => p = λ 1 The kinetic energy,

4 T = 1 mv => T = p m = mλ Consider a one dimensional wave function which describes the motion of a wave, ψ x = A sin π λ x (3) where A is the amplitude of the wave. Differentiating eq. (3) w.r.t x twice, we get d ψ dx = A(π λ ) sin π λ dψ dx = A π λ cos π λ x Substituting 1 λ = mt x => d ψ dx = 4π λ ψ(x) (4) from eq., we get d ψ dx = 8π mt ψ x 5 Also total energy of the system is, E=T + V(x) (6) eq. 5 => d ψ dx = 8π m E V x ψ x i. e., d ψ dx + 8π m E V x ψ x = 0 7 Eq. (7) represents one dimensional Schrodinger s time independent wave equation. b. de-broglie s hypothesis: According to de Broglie s theory, as light behaves as particle at one instant and as waves at other instant of time, every elementary like particle possess wave nature. As electromagnetic waves behave like particles, particles like electrons will behave like waves called matter waves. de- Broglie wavelength associated with a particle of mass m moving with velocity v is given by (or)

5 λ = mv (1) Based on Planck s theory of radiation, the energy of a photon (quantum) is given by E = ν = c λ () From Einstein s mass-energy relation, E = mc (3) From eq. () & (3): λ = mc = p (4) where p = mc is te momentum associated wit poton. If T is the kinetic energy of the material particle, then T = 1 mv = p => p = mt m Hence de-broglie wavelength, λ = / me (5) de- Broglie wavelength associated with electron(s): If electron of mass (m 0 ) and charge e being accelerated by potential V volts and v is the velocity attained by it then 1 m 0v = ev => v = ev m 0 λ = m 0 ev = 1.6 V Å (6) MODULE 3) a. Define the terms 1) Drift velocity ) Mean free path 3)Relaxation time 4) Resistance (4 Marks) c. State law of mass action. From this law obtain expression for Fermi level in an intrinsic semiconductor. (6 Marks) OR 4) a. What are the types of super conductors? Explain. (6 Marks) c. Explain in brief the construction and working of maglev. (4 Marks)

6 Soln.3: a. Drift Velocity: It is the average velocity with which the electrons are drifted in a direction opposite to the applied electric field. It is mathematically given as v d = J ne = σe ne Mean Free Path (λ): It is the average distance travelled by the electron between two successive collisions in the presence of applied field. λ = τc Relaxation Time (τ r ): It is the time taken for the drift velocity to decay to 1/e of its initial value after the electric field is switched off. Resistance: It is the physical property of the conductor which opposes the flow of current in it. According to Ohm s law, suc tat, V = IR Resistance, R = V I b. Law of Mass action: In a semiconductor the law of mass action states that the product of the electron and hole concentration is always equal to the square of the intrinsic carrier concentration, at a given temperature, i.e., n p = n i i. e., n i = 4 ( πkt )3 (m e. m ) 3 exp Fermi level in intrinsic semiconductor: In intrinsic semi conductors, no. of holes is equal to no. of electrons.the average energy of these charge carriers (electrons & holes) can be calculated and is called Fermi energy level (F) or E F and in intrinsic semiconductors, it is exactly middle of the valence band and conduction band n = p n = ( πm ekt )3/ exp ( E F E c KT ) = p = ( πm KT E g KT )3/ exp ( E v EF KT ) (m e )3/ exp ( E F E c KT ) = (m )3/ exp ( E v EF KT )

7 c. exp ( E F KT ) = ( m m )3/ exp ( E v+e c e KT ) by appling logarithms on both sides E F KT = 3 log ( m m ) + ( E v + E c ) e KT E F = 3KT 4 log m m e Assume = m m e = ( E v + E c KT ) E v + E c E F= ( ) Thus the Fermi level is located half way between the valence band and conduction band, and its position is independent of temperature. Since m > m e E F is just above the middle and rises slightly with increasing temperature. Soln. 4: a. Types of super conductor: Super conductors are classified into two categories depending on the way in which transition from super conducting state to normal states takes place under the presence of external magnetic field.(h c ) Type 1 superconductor: Materials in which a sudden transition from superconducting to normal conducting takes place are known as Type I super conductors. The magnetization curve drawn between magnetic field and magnetization for type-i superconductors as shown in fig (a).type I superconductors are perfect diamagnetic. Below H c it completely expel magnetic flux from the interior of the superconductor. Type II super conductors: The material in which gradual transition from super conducting state to normal state takes place is known as type II superconductors. In this case the magnetization curve is

8 shown in fig (b).for applied field below certain value H c1 known as lower critical field, the specimen completely behaves as superconductor and it is perfectly diamagnetic. At H c1 the field begins to penetrate the specimen and the penetration increases until H c is reached. At H c known as upper critical field, the magnetization vanishes and specimen becomes normal. A good type II superconductor excludes the field completely up to H c1 and partially in between H c1 & H c.the specimen remains electrically super conducting between the two fields. The value of H c may be 100 times more than lower H c1. b. Magnetic Levitation: (or Maglev or magnetic suspension) In this method an object is suspended with no support other than magnetic fields, a large DC current in a superconducting wire is used to generate high magnetic field and this result in superconducting magnets. In a super conductor the magnetization is in the direction opposite to that of the external applied field. This is called diamagnetism. When a superconductor is brought near a permanent magnet a strong repulsive force between the two, causes the lighter one to float over the other. This is known as magnetic levitation. Using the principle a magnetically levitated train is produced.the train floats without touching the (ground) tracks as a result of repulsion between its superconducting magnets and the magnetic field that they induce in the tracks. Magnetic levitation is used for maglev trains & magnetic bearings. & Magnetic television effect is used for high speed transportation.

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