Questions and Answers of Engineering Physics

                                 

1. Define Black Body?

      When heat radiations are incident on a surface, a part of it is reflected from the surface, another part is absorbed by it and the remaining part is transmitted through it.

       ie   r+a+t=100%           or  r+a+t=1

       r→ Amount of heat energy reflected

       a→ Amount of heat energy absorbed

      t→ Amount of heat energy transmitted

      A perfect blackbody is that which absorbs all heat radiations incident on it. There is neither reflection nor transmission of heat energy from that body.

r+a+t=1, generally for a perfect Black body, a=100%,      ie r=0, t=0

When black body is heated it emits all the heat radiations absorbed. The thermal radiations depend only on the temperature of the body 

2. What is a Ferry’s black body? Explain the black body spectrum?   (CU 2006)

      It is a double walled Sphere with a very fine hole ‘O’ at one end a projection P on its opposite inner side. The inner surface is perfectly coated with lamp black. The projection ‘P’ prevents the radiations from direct reflection. The inner space between the walls is evacuated

When heat radiations are incident through the hole ‘O’ they undergo multiple reflections until they are completely absorbed by the inner surface. Now it become a perfect black body absorber. When this body is heated to a temperature it emits full radiations through the hole. The internal radiations depend only on the temperature of the body

Distribution Of Energy In The Spectrum Of A Black Body

      The distribution of energy of radiations from a black body was first analyzed by Lammer and Pringsheim. The heat radiations passed through carbon tube to slit “s’(they emitted from Carbon tube).The beam falls on a Concave mirror(reflector). The II ’l beam is now passed through a flours per prism which disperses it. The dispersed beam is focused to a Bolometer which measures the intensity (energy) of thermal radiations. By rotating the prism the energies corresponding to different wave at a   particular temperature are measured it. The measurements are made with thermal radiations of different temperature.  A graph is plotted with the wavelength on x axis and the intensity of relation of radiation E on y axis

reaches maximum value E_m

From graph, the energy is not distributed uniformly over the spectrum. At a particular temperature the intensity of radiation E (energy) at first increases with wavelength  and then decreases with further increases of wavelength. The particular wavelength corresponding to the maximum intensity of emission is taken as m. Increases of temperature produces increase in energy of all wavelengths. When temperature of black body increases Emission of maximum energy at a particular also gets increased. As temperature increases the wavelength ‘(particular    for maximum energy emission) corresponding to the maximum emission of energy gets shifted towards the shorter wavelength side ie _m gets decreased. This is represented in dotted lines

         ie     ∝1/T

       mT= a constant .It is called Wien’s law

The area between the curve and the wavelength axis gives the total energy emitted per sec per unit area of the blackbody at the temperature. It is found that this area (ie energy) directly proportioned to the fourth power of its absolute temperature

            ie E∝T⁴

it is called Stefan’s Law

3. What are matter waves and De Broglie waves?           (KU MAY 2005)

In the phenomenon of interference and diffraction light behaves as a wave while in photoelectric effect and Compton effect it shows particle nature. Thus light has a dual nature. De Broglie hypothesis says that every moving matter exhibit wave like properties under suitable conditions. The wave associated with a particle is called a matter wave.

4. Give an expression for of Broglie wavelength of an electron

Consider a photon of mass m and momentum ‘p’ and frequency ‘’.

Momentum is given by (p=mc)
 
 
                                                                                                     

      De Broglie wavelength

Wavelength of Electrons

Consider an electron of mass ‘m’ and charge e subjected to a potential difference of V volts. If ‘V’ is the velocity acquired by the electron

then ½ mv² = eV                 

   

Substituting values, of h, m and e, we have

                                                                                                  

Wavelength of electron wave is inversely proportional to the square root of the accelerating potential. The wavelength of de Broglie waves associated with electrons accelerated through a potential difference of 100 volt  in vacuum is 1.23 A⁰. This value is of the order of the wavelength of x- rays. Davisson and Germer proved experimentally the wave nature of electrons by electron diffraction experiment.

5. What are wave Packets?

      A wave that is confined to a small region space in the vicinity of the particle is called a wave packet. It is an envelope of a number of waves super imposed.

by de Broglie hypothesis

      We can use wave equations to describe the small particles. This is the significance of de Broglie equations.ie the character of matter waves can be explained with the idea of wave functions.

6. Explain Heisenberg’s uncertainty principle?  (KU MAY 2006, CU 2009,2006,2005)

      By classical mechanics the position and momentum of a particle can be determined simultaneously with accuracy. In Newtonian mechanics every particle has a fixed position in space and has a definite momentum at any time. According to uncertainty or the principle of indeterminacy.  “It is impossible to have an accurate measurement of the position and momentum of particles simultaneously”. The product of the uncertainty in the measurement of position of the particle at a certain instant and the uncertainty in the measurement of the momentum of the particle is of the order of the Planck’s constant ’h’.. If  ∆x is the uncertainty (error) in the measurement of the position of particle along x coordinate and ∆Px is the uncertainty (error) in the measurement of its momentum, then 

                          ∆_x∙∆P_x= 

            Similarly    

                                  

ie if  is small., is large and vice versa.

If =0,     = infinity

Also     = infinity

        ∆E.∆t=
                                                                                  

      An electron exists in an excited state only for a short interval of time. Thus ∆t is small, ∆E must be large

            The typical value of ∆t=10^-8seconds.

Also    E=h     ∆E=h∆                        ∴ ∆=  but ∆E=           

     

                                                                                            

This is the irreducible limit to the accuracy with which we can determine the frequency of radiation emitted by an atom which remains in the excited state for about ∆t=10^-8 seconds.

7.  What are ultrasonic waves? Mention their properties.     (CU 2009, 2008)

Ultrasonic wave means acoustic wanes whose frequency is greater than 20 kH_z. This is a very powerful tool in non-destructive testing of materials, structures and products in engineering.

Ultrasonics

      Acoustics is a branch of physics that deals with the study of mechanical waves which gives rise to sound. Sound waves are longitudinal mechanical waves. They can propagate in solids, liquids and gases. Sound waves in the frequency range which can stimulate human ear and brain to the sensation of hearing (ie 20 HZ to 20 KHZ) is called audible range.

Below audible range - infrasonics →eg: earth quake, elephant sound, etc

Above audible range → ultrasonics -eg: elastic vibrations produced by quartz crystal, sound produced by bats etc

 Properties of Ultrasonics:-

1.     They cannot travel through vacuum

2.      They are high energetic waves

3.      The speed of ultrasonic waves in a thin rod or crystal is given by

              v=

              Y→ Young’s modulus of the material

              ρ→ density of the rod

Its speed in liquid is given by

                                       K→ Bulk modulus

                                                          ρ→ its density

      In gas,. 

      v=                                                   P→ pressure of the gas

                                                                          ρ→ its density

4.      Speed of ultrasonic waves depends on frequency, greater frequency higher speed.

5.      They can be reflected, refracted and diffracted like light waves.

6.      When ultrasonic waves travel through a medium, they are scattered and a part of energy is                  absorbed by the medium. This loss of energy by scattering and absorption is called attenuation.

7.      They produces heating effect when pass through a medium. A part of energy absorbed by   the medium reappears as heat energy

8.      When ultrasonic waves pass through a liquid, stationary waves are produced by the reflection. As a result the density of liquid layer is varied from layer to layer. This behaves like a plane diffracting grating which can diffract light waves.

9.      Stirring effect

Intense ultrasonic beam produces vigorous agitation in certain low viscous liquids. They produce bubbles due to the disruptive effect.

8.   Define piezoelectric effect? What are all the methods available to produce Ultrasonic waves? Explain piezoelectric method. (CU 2005, 2008, 2009)

      When certain crystals like quartz tourmaline etc are subjected to stress or pressure along certain axis, a potential difference is developed across the perpendicular axis. This is called Piezo electric effect, discovered by J-Curie and P- Curie in 1880. The converse of this effect is also possible  when a potential difference is applied between the two opposite faces of a crystal than stress or strain is induced in the other two opposite faces, ie the crystal is set into vibrations. If the frequency of elastic oscillations coincides with the natural frequency of the crystal, the vibrations will have large amplitude. The crystals which exhibit Piezo- electric effect are called Piezo electric crystals.

      E: Quartz, tourmaline, Rochelle selt etc.

  `There are 3 main methods available to produce ultrasonic waves

1.      Mechanical generator

2.      Piezo electric effect method

3.      Magneto striction method

Piezoelectric generator Construction

      It is based on the converge of Piezo electric effect

      It consists a quartz crystal placed in between the parallel metal plates and it is connected parallel to a tank circuit. The inductance coil L₁ and L₂ is centrally tapped. The coil and the variable capacitor constitutes the tank circuit. One end of the tank circuit is connected to the base of the transistor. The electric power required to operate the transistor is supplied from a D.C source V_CC through R.F.choke. R.F Choke prevents a.c. from flowing to dc source. R_E is a resistor connected to emitter and it regulates the potential of the emitter. C_E is an A.C bypass capacitor which by passes ac alone. R₁ and R₂ resistors provide proper voltage to the transistor. The coupling capacitor C_in prevents the dc flowing to the transistor.

      WORKING

      When key is switched on high frequency ac is produced from the tank circuit. When this ac is applied across a part of opposite faces of the quartz crystal, it begins to vibrate according to the principle of Piezo electric effect. The variable capacitor is tuned so that the applied frequency becomes equal to the natural frequency of the crystal. Hence tuned resonance takes place and the crystal vibrates with the maximum amplitude. At this condition ultrasonic waves are produced from the sides of the crystal. Ultrasonic waves up to 15 MH_Z can be produced using this method.

      The frequency of the circuit is given by

        If        L= L₁ +L₂ then                        

      Natural frequency of the crystal is given

                                                                            Y→ young’s modulus of the crystal

                                                                                  l → thickness of the crystal

                                                                                 ρ → its density

9. Define magnetostriction. Explain this method to produce ultrasonics? (CU 2005, 2011)

      When a ferromagnetic material in the form of a bar is subjected to an alternating magnetic field with its length parallel to the magnetic field as shown in fig., the bar undergoes alternate contractions and expansions at a frequency equal to the frequency of the applied magnetic field. This is known as magnetostriction effect.

 

                               

                       

      Ferromagnetic materials which are used for the production of ultrasonic waves are called magnetostriction materials.

PRODUCTION OF ULTRASONICS BY MAGNETOSTRICTION

      The transistorized version of the magnetostriction oscillator is shown in the figure. R is a ferromagnetic rod clamped at the middle. A coil is wound on R near one end and insulated from it. The coil L₁ and variable capacitor constitution the tank circuit. The frequency of the tank circuit is equal to the fundamental frequency of longitudinal vibrations of the rod R. The tank circuit is connected to the collector of the transistor T. The other end of the rod carrier another coil L₂ wound on it. L₂ is connected to the base of the Transistor through the coupling capacitor Cin. The coils are wound loosely on the rod so that rod can vibrate freely

      WORKING

      When the key k is closed a current flows through the coil L₁. Then a magnetic field is produced. This magnetic field changes the length of the rod due to magnetostriction. This magnetic field changes the length of the rod due to changes the magnetic flux through L₂ thus inducing an emf in it. The coil L₂ is connected between the base and the emitter of the transistor. Hence the emf across it will increase the forward bias.

      ∴ The collector current increases. This provides a +ve feedback which is required  to sustain the oscillations of the tank circuit. The voltage across the Resistors R₁ and R₂ provide proper biasing of the transistor. C_E is ac bypass capacitor. The Magnetostriction method can be used to produce ultrasonic of frequency range 5KHZ to 60 KHZ.

      The applied frequency is given by,

      Natural frequency of the crystal,

                                                            Y → young’s modulus of the crystal

                                                                        ρ→ its density 

      Disadvantages of magnetostriction method.                      

      Magnetostriction method is somewhat expensive. Low range frequency ultrasonic can be produced alone up to 60 KHZ. Efficiency affected by eddy current loss and hypothesis loss

      These are the disadvantages of this method

10. Mention the Applications of ultrasonics? (CU 2009)

      1).Non Destructive Testing of materials (NDT) 

      It is a method of testing of materials without any destruction in the materials. The present condition and quality of the materials can be examined using NDT without destroying their properties. Ultrasonics can be used to detect the imperfections like the flaws, cracks, breakings, cavity,  air pockets, discontinuities etc in material like metals. Any defects in weldings and castings can be exactly located with the help of ultrasonic. Aircraft have to undergo such ultrasonic testings before their flight.

      A flaw or a crack in metal block produces a change in medium which causes reflection of waves. When ultrasonic waves pass through a flaw it is partially reflected from it and partially transmitted through it. The transmitted beam is again reflected from the bottom side of the metal block. When ultrasonic beam is incident on the interface between two media it is partially reflected and partially transmitted. The intensity of the reflected and the transmitted beam is decided by the acoustic impedences of the media. This is the principle of ultrasonic testing. The intensity of ultrasonic wave from the cavity examined using a CRO

                       →intensity shown by CRO if the object having no cavity or breakings

 

                   → Intensity graph shown by the CRO suppose the object having cavity or breakings etc                    

     2) SONAR

      In this method, high frequency ultrasonic waves are used to find the distance and direction of submarines, depth of sea, depth of rocks in the sea shoal of fish in the sea etc. The ultrasonic waves are generated by the Piezo electric method using a quartz crystal placed in between two metal plates. The same quartz crystal is used to detect the ultrasonic waves.

 

 

 

                      

 

Bed of sea

      These waves are transmitted towards the bed of the sea and get reflected back from the bed in the form of echos.  These reflected echos are received by the quartz crystal and they are amplified and fed to the CRO. The time taken by the ultrasonic waves for the to and fro travel is measured

d=

           
                             

      d→ depth sea

      t → time taken for to and fro travel

      v→ velocity of ultrasonic waves

      3) Ultrasonic waves are very effective in cleaning material surfaces. It will agitate dust and   impurities on surfaces and remove them.

      4) Ultrasonic drilling and welding are very effective in drilling holes and to weld soft metals and plastics.

      5) Ultrasonic waves can accelerate chemical reactions. The technique is used by chemical industry to reduce reaction times.

      6) They are used for Sterline milk, water etc.

      7) Unicellular organisms can be destroyed when exposed to Ultrasonics.

      8) Diagnosis is mostly based on Ultrasonic Scanning and imaging.

      9) The ultrasonic waves can be for directional signaling on account of their high frequency. It can be concentrated into a sharp beam and can be used for signaling in a particular direction.

      10) We can use ultrasonic waves to find the velocity of sound in gases and liquids.

11. Define reverberation & reverberation time? (CU 2008)

      The sound produced in a room or a hall suffers multiple reflections from various objects like the walls, ceiling, floor, furniture etc in the hall. The sound appears to remain for a long time after the source of sound is stopped. This persistence of sound even after the sources of sound is stopped is called reverberation.

      The reverberation time (T) is defined as the time taken for a sound to decreases in intensity to10^-6 of its original intensity, the time being reckoned from the instant when the source of sound is cut off. The time of reverberation is also defined as the time required for the intensity of sound to decrease by 60db from the moment when the source is cut off. The reverberation time is highly significant in the design of halls and auditoriums.

12.  Define absorption coefficient?

      The absorption coefficient (∝) of the surface of a material is defined as the ratio of the sound energy absorbed by the surface to the total energy incident in it. Since an open window absorbs the whole amount of sound energy incident on it, the absorption coefficient for it is unity. The unit of absorption is called Sabine. It is the sound energy absorbed by on square Foot of an open window. For unit area of various surfaces, the absorption coefficient is expressed in terms of equivalent area of open window. The equivalent absorbing area A for a surface having total area S and absorption coefficient ∝ is given by

               A=∝S

13. Define

(i)     Matter waves

(ii)   Wave packets

  

 i) Matter waves

       In the phenomenon of interference and diffraction light behaves as a wave while in photoelectric effect and Compton effect it shows particle nature. Thus light has a dual nature. De Broglie hypothesis says that every moving matter exhibit wave like properties under suitable conditions. The wave associated with a particle is called a matter wave

ii) Wave packets

      According to de Broglie hypothesis a wave is associated with a moving particle. Hence a particle can be represented by a wave confined a space. A plane wave cannot be used to represent ot since it extends to infinity. A wave that is confined to a small region space in the vicinity of the particle is called a wave packet. It is an envelope of a number of waves super imposed

14. Explain the physical concept of wave function (CU 2011)

      The quantity with which quantum mechanics is concerned is the wave function ψ of a particle. The quantity that undergoes periodic changes of a body is called wave function ψ. It is in general a complex valued function and itself has no physical interpretation. The square of the absolute magnitude / ψ or ψ dxdydz is proportional to the probability of finding the particle in  the small volume element dxdydz about the point x, y, z. we can obtain all the physical properties of the system if we know the wave function.

      The wave function should fulfill certain requirements.

      Since ψdxdydz is proportional to the probability of finding the particle with in the volume element, the integral ψ dxdydz must be finite if the particle is somewhere there. If ψdxdydz is zero, the particle doesn’t exists and if it is infinity, the particle is everywhere simultaneously.
Since the probability of finding the particle in the volume element is a surety, then ψ dxdydz must be equal to 1. The wave function satisfying above condition is called normalised wave function.

     The requirements of wave function

1.      The wave function ψ must be continuous and single valued everywhere

2.       ,  and  must also be continuous and single valued everywhere

3.      can be normalized.

15. Derive Schrodinger’s Wave Equation for a free Particle and time Dependent Equation      (CU 2010)

      Schrodinger’s equation is the basic expression used in quantum mechanics. This cannot be derived from elementary rules. We can derive it by considering the plane wave equation and combining with Einstein’s equation for quantum of energy and de Broglie’s’ expression for wavelength.

      A particle in motion is associated with a wave function that contains the information about the motion. A plane progressive wave that propagates along X - direction is given by

              Ψ = A‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ (1)

      Where k is the wave vector given by and  is the angular frequency.

      Einstein’s formula for photon energy is

              E= h v=   = where h =

      de – Broglie’s expression for matter wavelength is  =

               p = =  .  = hk

      Using the above expression, equation (1) becomes

                        Ψ = A

                        Ie ψ = A 

      On partial differentiation of Ψ with respect to x, twice, we get

               =

              Differentiating with respect to time,

               =                 

              These are equivalent to

              ψ = -‑‑‑‑‑‑‑‑‑‑‑‑‑ (4)

              = - i( - i ) ψ   ‑‑‑‑‑‑‑‑‑‑(4a)

              From 3 and 4, p = i

                        This is called space operator.

                                    E  = i‑‑‑‑‑‑‑‑‑‑‑‑ (5)

                                    i‑‑‑‑‑‑‑‑‑‑‑ (5a)

              This energy is called time operator.

              For a free particle total energy is given by

                                    E =  , since V = 0

                                    ie E  = ‑‑‑‑‑‑‑‑‑‑ (6)

      using equation (4) and (5), Equation (6) becomes

                         = iħ‑‑‑‑‑‑‑‑‑‑‑‑ (7)

      This is Schrodinger’s equation for a free particle in one dimension. In three dimension it becomes for a free particle, as

                        ψ = iħ‑‑‑‑‑‑‑‑(8)

                        Where  =  +  +

      Is called Lapalcian operator

      If the particle is moving under a potential V (r, t) then equation (8) becomes

               = iħ

      This is Schrodinger’s time dependent equation

16. Derive the time independent Schrodinger wave equation or steady state equation. (CU 2010)

      According to Schrodinger, de- Broglie’s wavelength holds good for any particle moving in any field of force with potential energy v.

     Then total energy E = kinetic energy + potential Energy

              E =  m  + V

                  =  + V

              ie = 2 m (E – V)

      or         p = [ 2m (E – V)

     The wave equation in Cartesian co-ordinate system can be written as

      +  +  = ‑‑‑‑‑‑‑‑‑‑‑‑ (9)

      Where  = ‑‑‑‑‑‑‑‑‑‑ (10)

     ‘u’ is the velocity of motion and  (x y z t) represents the amplitude of the wae associated with the particle

      From (10), we get

               = -

     So Equation (9) becomes

       +  =  ψ ‑‑‑‑‑‑‑‑‑‑‑‑‑ (11)

     

       =  =

     So we have

     

      +  =  ψ or

              ψ +  ψ = 0 ‑‑‑‑‑‑‑‑‑‑‑‑‑ (12)

      Equation (12) is a general equation which is independent of time. Let us now introduce the concept of de-Broglie wavelength.

                                     =  =

                                                  =

      Substituting the wavelength is equation (12)

                                    ψ +  [ψ = 0

                                                            or

                                    ψ + ψ = 0

                                                            Or

                                    ψ + ψ = 0………………. (13)

              This represents Schrodinger’s time – independent wave equation.

              Equation 13 shows that the wave function ψ is a function of coordinates also. V is a function of coordinates.

      For the case of a free particle (V=0), the Schrodinger equation becomes

                        ψ + ψ = 0

      E is the energy having definite values and it also has to be satisfied with certain boundary conditions. The discrete values of E are called Eigen values and the correspondence wave functions are called Eigen functions.

17.  Define Expectation values

     In quantum mechanics each dynamic variables is represented by an operator which acts on a wave function to give a new wave function.

      The expectation value of an operator ‘A’ representing a dynamic variables, denoted by , is defined as

      Consider a large number of identical systems. They are in the same state of wave function ψ before measurements. Expectation value is the mean or average value of the results obtained by the measurements. Dynamic quantities like position, momentum, energy etc. are called observables. In quantum mechanics each observable is represented by an operator. When an operator is acting on a wave function we get a new wave function.

              < A >=

      Where d r is the differential volume element. If ψ is a normalized wave function, then

               = 1, then

      < A >=      

     Expectation value depends on the state of the system before measurement. It is the mean value of he results obtained by performing the measurement on a large number of identical systems each of which v was in the same state  before measurement. To emphasise the fact, one may write the expectation value as

18. Derive the time independent Schrodinger equation and solutions

      In time dependent Schrodinger equation, the potential energy of a moving particle is a function of time and position also. In certain cases potential energy does not depend explicitly on time. Then we get time independent Schrodinger equation for such particle or system.

      The wave function in this case can be expressed as a product of two functions. (r), a function of position only and f(t), a function of time only

              Ie

              Then  =

                        and= f(t) ψ

     putting this on time dependent Schrodinger equation,

       +  =  (r, t)

                        f(t) + f(t) = iħ

              Dividing throughout by ψ (r) f(r)

                         +  =  ……… (14)

      In the above equation, the variables are separated.

      LHS is a function of position only and the RHS is a function of time only. Therefore each side must be equal to a common constant called separation constant.

              Ie = E

               = dt.

     On integration

                        Log f =  + constant.

                        Or f = C ‑‑‑‑‑‑‑‑‑‑‑ (15)

      Equating the LHS of equation 14

               +  = E

               = E – V  or

               + ( E – V) = 0 ‑‑‑‑‑‑‑‑‑ (16)

      The above equation is the Schrodinger time independent wave equation.

              This is applicable to problem with potential energy independent of time. This will be applicable to steady state or stationary state problem.

19.  Solve the Schrodinger equation for a particle confined in a one-dimensional box of length L. Draw the first few energy levels and the corresponding eigenfunctions. (CU 2011)

      Consider the motion of a particle of mass m confined to move between two walls of infinite height at x = 0 and x = L. The width of the box is L. Since there is no interaction between the particle and box, the potential energy of the particle is taken to be zero. It is very clear that potential energy of the particle is taken to be zero. It is very clear that potential energy does not depend on time and we are considering only time independent. Schrodinger equation for the solution. The problem is one dimensional and the Schrodinger equation is.

                         +  (E – V)ψ = 0

                                    Since V = 0, the equation becomes

                         +  ψ = 0    or

       +ψ = 0         where  = ‑‑‑‑‑‑‑‑‑‑‑‑‑ (17)

      The solution of the above differential equation is of the form

              Ψ = A sin kx + B coskx‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑( 18)

      The solution has to be well behaved wave function. Since the particle is inside a box of infinite height, it is impossible to find the particle outside the box ie ψ must be zero for all points outside the box

              Ψ = 0 for x < 0 and

              Ψ = 0 for x > L

      This is possible only if = 0 at x = 0 and x = L as demanded by continuity condition.

 

                                          V = 0

                                               

                               X = 0        X = L                     X

      Applying first condition on equation 18, we get

                        0 = A sin 0 + B cos 0

              ie B = 0

      So solution reduces to

              Ψ = A sin Kx‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ (19)

      Using the next condition ψ = 0 at x = L, we get

              0 = A sin kL

      There are two possibilities. Either A = 0 or sin kL = 0

      A cannot be zero, since the wave function cannot exist. So we have the other possibility

              Sin kL = 0

              ie kL = n‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ (21)

      If n = 0, then ψ = 0 for all values if x. This is ruled out. Therefore n is a non-zero integer. We have  =  on using equation (17)

       =                     or

       =  . ie=

            Since ħ =  as the energy corresponding to n the different values of energy for n are called Eigen values. Since n is restricted, the particle cannot have any value of energy, but restricted to certain values. The quantity n is called quantum number.

Eigen Function

     By applying the normalization condition, ie ∫ψ d r= 1 we can find the normalized wave function. ∫dr =1

      So we have A sin  A sin  dx = 1

              dx = 1

                        dx = 1

      On integration and applying limits we get

                         = 1                                    or         A=

      So the normalized wave function is

               (x)=  sin  where n = , 2, 3 ‑‑‑‑‑‑‑‑‑‑‑‑‑‑‑ (22)

      For the ground state n = 1, the wave function is given bt

                        =  sin

      Similarly the wave function for the first two excited states are given by

                        =  sin  and =  sin

              These wave functions associated with different energy are called Eigen wave functions.

             

 

     

                                Ψ                                      

                                                                   

                                                                       

     =0                                      
                                                   X=L

                         wave function of first three energy levels

20. How will determine the velocity of ultrasonic waves in a liquid by ultrasonic diffractometer?

                                                                                                                                    (CU 2008)

Velocity and wavelength of ultrasonics using ultrasonic diffractometer

When a quartz crystal Q placed between two metal plates is a liquid is set into vibrations using an R.F. Oscillator, ultrasonics are produced. When these ultrasonics are reflected by a reflector, longitudinal stationary waves are produced in the liquid. As a result alternate nodal pulses and antinodal pulses are formed. At nodal planes, the layers are crowded together (compressions or condensations) and density is maximum. At antinodal planes, the layers are separated (rarefactions) and density is minimum. This setup of  nodal planes and antinodal planes behaves like slits and opaque spaces of a plane grating. Such an arrangement is called acoustic grating. Using this acoustic grating, velocity 'V’ and wavelength  of ultrasonics can be determined.

A parallel beam of monochromatic light from a sodium vapour lamp is collimated and is allowed to is observed fall normally on this acoustic grating. Diffraction takes place and the diffracted beam through the telescope of a spectrometer. On either side of the central maximum various orders of principal maxima are obtained. If θ is the angle of diffraction for a principal maximum, then

d sinθ = n  ———> (1) where'd' is the distance between two consecutive nodal planes or two consecutive anti nodal planes, n is the order of spectrum and  is the wavelength of monochromatic light d can be calculated from this grating equation. But

      d = _a/2  or   _a = 2d---------- >(2) where _a is

      the wavelength of ultrasonic wave in the liquid.

But V=. Where  is the frequency of oscillations of the crystal and V is the velocity of ultrasonic wave in the liquid.

DESCRIPTION

Ultrasonic diffractometer mainly consists of a quartz crystal Q placed between two metal plates provided with connecting leads. The quartz crystal setup is clamped inside on one face of an optically plane rectangular glass cell filled with kerosene or CCl₄ so that the crystal is immersed completely in liquid. The cell is placed on the prism table of a spectrometer and it is illuminated by a beam of monochromatic light from a sodium vapour lamp. The quartz crystal can be subjected to vibrations from an R.F.oscilIator.

PROCEDURE

The initial adjustments of a spectrometer are made. The rectangular cell containing liquid is placed on the prism table perpendicular to the collimator. The quartz crystal Q together with metal plates and connection leads is clamped inside the liquid on another side of cell. A narrow beam of monochromatic light from collimator is allowed to fall normally on the cell. The direct image is observed through the telescope. Now the R.F. oscillator is switched on and the frequency of R.F. oscillator is varied from 2 to 5 MHz . The crystal is subjected to these oscillation and it begins to vibrate resonance with the oscillator. As a result ultrasonic waves are produced in liquid. These waves get reflected from the opposite side of the cell producing longitudinal stationary waves in the form of compressions and rarefactions. This arrangement behaves like a grating and as a result ultrasonic waves are diffracted. On either side of central maximum different orders of principal maxima are obtained. The telescope is turned so that the cross-wire coincides with the spectral line of first order on one side of the central maximum. The main scale reading and vernier scale reading are noted. Now the telescope is turned to the other side so that the cross wire coincides with the spectral line of first order. Main scale reading and vernier scale reading are noted. From these two sets of readings 2θ  for first order and hence θ  are calculated. The distance'd' between two consecutive nodes or antinodes is found out from the' grating equation dsin θ =n where  the wavelength of sodium light is known (5893Å) _a the wavelength of ultrasonics in liquid can be found out from _a = 2d. Knowing the frequency of oscillator, velocity V of ultrasonics in liquid is calculated from the equation, V = _a.

21. Derive Sabine’s formula for reverberation time? (CU2008, 2011)

      Sabine derived an expression for the reverberation time (T) on the basis of the following assumptions.

·         The distribution of sound energy and the intensity of sound is uniform inside an enclosure.

·         The dissipation of energy in air is negligible.

·         The absorption coefficient of any surface is independent of the intensity of sound.

·         The phenomenon of interference and formation of stationery waves are supposed to be absent or non- existent and

·         The rate of emission of sound energy is constant.

      Consider a hall of volume V. Let a source of sound emit sound. The sound energy spreads out in all directions. The sound waves (energy) are partially reflected and absorbed by various objects in the hall. After some times, a steady state is reached between the energy emitted and the energy dissipated. Then the energy density (energy/ unit volume) becomes uniform throughout the hall. 

      Let the source of sound be cut off at =0. Let E₀ be the energy density at this instant. The energy density decreases exponent all with time. Let E be the energy density after secs. Then

      E= E₀e^-Avt/4V  ------(1)

      A= the total energy absorbed

      V= the velocity of sound

      V= the volume of the hall

      For a given frequency of sound, the intensity of sound is proportional to the energy. Hence if I₀ is the intensity at t=0 and I is the intensity after t secs

      I=I₀e^-Avt/4v 

When t= T, the reverberation time,

       (From definition of reverberation time)
        Then from equations (2) and(3)

Taking logarithms,

      

        =2.303×log₁₀10⁶

        =2.303×6×1

Taking =340m/s at root temperature

                


 This is Sabine’s formula for reverberations time

              A=              

The total energy absorbed by various surfaces

22. Define the terms

(i)     Nano science and Nanotechnology

(ii)    Nano materials  (CU 2010)

(iii)  Nano clusters

(iv)  Fullerenes 

(i) Nano Science and Nanotechnology

      Nanotechnology is the study of the control of matter on an atomic and molecular scale. Generally nanotechnology deals with structure of the size 100 nanometer or smaller, and involves developing materials or devices within that size. Nanotechnology is very diverse, ranging from novel extension of conventional device physics, to completely new approach based upon molecular self assembly, to developing new materials with dimensions on the nanoscale, even to speculation on whether we can directly control matter on the atomic scale. One nanometer (nm) is one billionth or 10^-9, of a meter.          

(ii) Nano materials

Nano materials are the materials of a size one billion or 10^-9, of a meter. Nanotechnology is very diverse, ranging from novel extensions of conventional device physics, to completely new approach based up on molecular assembly, to develop new materials with dimensions on the nano scale. Molecules of the dimension 0.1nm. Most of the atoms are on the surface of the clusters. The size of the particle is less than the critical characteristics length of the electron to conduct, they exhibit different properties. 

(iii) Nanoclusters

     

Clusters belong to a new category of materials. Their size is in between bulk materials and their atoms or molecules. Their properties are fundamentally different from those of discrete molecules and bulk solids. They are systems of bound atoms or molecule existing as an intermediate form of matter with properties that lie between those of atoms and bulk materials. Depending on the constituent units they are called either atomic or molecular clusters J Clusters include species only in the gas phase or in the condensed phase or both. They can have either a net charge (ionic clusters) or no charge at all (neutral clusters). The atoms or molecules constitute clusters are bound by forces which may be metallic, covalent, ionic hydrogen bonded or van der. Waal's in character and can up to a few thousand atoms.

iv)    Fullerenes

      Fullerenes are molecular forms of carbon which are distinctly different from the extended carbon forms known for millennia. There are numerous forms all of which are spheroidal in structures. In Chemistry there is no other molecule formed by the same atom which is as big as fullerenes. A carbon molecule with chemical formula C₆₀ containing 60 carbon atoms in the shape of a soccer ball had been predicted in 1970. An experiment was carried out in Rice university in which a graphite disc was heated by a high intensity laser beam that produces a hot vapour of carbon. A burst of helium gas then swept the vapour of carbon out through an opening where the beam expands. The expansion cooled the atoms and they condensed into clusters. This cooled clusters beam was then narrowed by a skimmer and fed into a mass spectrometer to measure the mass of the molecule in the clusters. A mass number of 720 that would consist of 60 carbon atoms, each of mass 12 was seen. This was evidence of a C₆₀ molecule. This was named after the architect Buckminister Fuller. The name Buckminister fullerene was shortened to fullerene.

23. Mention the applications of Nanotechnology?

Nano medicine: Medical research field has exploited the unique properties of nano materials for various applications, such as cell imaging and treating cancer. This field is called nano medicine. Nanotechnology is also used for diagnostic purposes. The drug consumption and side-effects can be reduced significantly by a targeted or personalized medicine and hence the treating expense is lowered. This is done with the help of nanotechnology by depositing the medicine in the morbid region only and in the correct doze. Nanotechnology can help to reproduce or repair damaged tissue.

Energy: The applications of nanotechnology in the field of energy are

1.   by reduction of energy consumption

2.   By increasing the efficiency of energy production.

3.   By the use of more environmentally friendly energy systems and

4.   By recycling batteries

Nanotechnological approaches like LEDS or Quantum Caged Atoms (QCAs) could lead to a strong reduction in energy consumptions. The efficiency of internal combustion engines could improve combustion by designing specific catalysts with maximized surface area. An eg. for an environmentally friendly form of energy is the use of fuel cells powered by hydrogen which is produced by renewable energies. The use of rechargeable batteries with higher rate of recharging using nanomaterials could be helpful for battery disposal problem  

Chemistry and Environment: Chemical catalysis from nano particles is highly beneficial due to its extremely large surface to volume ratio. The application ranges from fuel cell to photo catalytic devices.

Filtration: Nanofiltration helps waste water treatment, air purification and energy storage devices.

Information and communication: Memory storage devices such Nino-RAM is developed using carbon nanotubes based crossbar memory.

Food: New consumer products created through nanotechnology are coming to market now. There are nearly a thousand known or claimed nanoproducts. The eg. of food products are canola cooking oil and a tea called nanotea etc.

Textiles: The use of engineered nanofibers already makes cloths stain –repellent and wrinkle-free. These textiles can be washed less frequently and at lower temperatures.

24. Explain the properties & applications of carbon Nano tubes?

Carbon nano tubes are ultimate high strength carbon fibers. They have a high strength to weight ratio. This value is 100 times that a steel. They are highly resistant to chemical attack. It is difficult to them. As a result temperature is not a limitation in practical applications of nano tubes. Surface are of nano tubes is higher than that of graphite. Nano tubes have a high thermal conductivity exhibit a striking telescope property.

      In multi walled Nano tubes, multiple concentric nano tubes Presley rested within on another exhibit a striking telescoping property. This is one of the first example of molecular nanotechnology; the precisely positioning of atoms to create useful machines. Thus property has been utilized to create world’s smallest rotational motor.

      Because of the symmetry and unique electronic structure of graphine, Nano tubes can be metallic with 1000 times’ greater conductivity then metals or moderate semiconductors. Because of this property CNT are referred to as “one dimensional”

Applications of CNT

Nano tubes can be used as very good electrical conductors. An application of nanotubes is the production of CNT based field emission displays. CNT act  as electron emitters  at lower turn on voltage and high emissivity. Nanotube tips can be used as a nanoprobs, which does not crash frequently. It can be used for making paper batteries. It is a paper thin sheet of cellulose infused with aligned carbon nanotubes. A CNT complex formed by CNT & fullerenes are used as solar cells. CNT have been implemented in nanoelectromechanical system like nanomotors. Nanotubes are used to form alter capacitors.