Amity School of Engineering & Technology (NOIDA)

Applied Physics I

Tutorial Sheet: 1 (Module I:Oscillations and Waves )

1. A simple harmonic oscillator of mass 0.4 gm has amplitude of 2 cm and the velocity at zero displacement is 100 cm/s. Calculate the periodic time of oscillation and the energy of the oscillator.

2. A particle vibrates with SHM of amplitude 0.05 m and a period of 6s. How long does it take to move from one end of its path to a position 0.025 m from the equilibrium position on the same side?

3. A particle is moving under SHM .When the distance of the particle from the equilibrium postion has the values x and x ,the corresponding values of velocities are v and v .Show the time period is:

T=2π√x - x / v - v

4. Show that if the displacement of a moving point at any time is given by an

equation of the form x=acosωt+bsinωt, the motion is simple harmonic. If

a=3, b=4 and ω=2, determine its period, amplitude, maximum velocity and

maximum acceleration of the motion.

5. The general solution to the equation of simple harmonic oscillation,

is x(t)= where A and C are constants. Show that

this can also be written in the form x (t)= and x(t)=

Find out the expressions .

6. String on a guitar has a linear mass density of 3 gm/m and is 63

cm long. It is tuned to have a fundamental frequency of 196

Hz.

(a) What is the tension in the tuned string? (b) Calculate the wavelengths of the first three harmonics. Sketch the

transverse displacement of the string as a function of x for each of.these harmonics

7. A bell is ringing inside of a sealed glass jar that is connected to a

vacuum pump .I nitialy ,the jar is filled with air .

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The air is slowely removed from the jar by the pump ? Describe the effect on following properties

(a).The sound intensity

(b).The frequency of the sound.

(c)The speed of the sound

8. Discuss the characteristics of simple harmonic oscillators. What is a quality

factor and how do you define it?

9. A mass of 1.6 kg stretches a spring 0.08 m from its equilibrium position.

This mass is replaces by another mass of 0.05 kg. It is pulled and released

so that it starts oscillating. Find the time period of oscillation.

Tutorial Sheet :2 (Module I:Oscillations and Waves)

1. If a particle is subjected to a restoring force proportional to displacement and a damping force proportional to velocity, calculate the displacement of the particle at any subsequent time it is executes oscillatory motion. If the motion of the particle is critically damped, find the displacement at any instant.

2. For a damped harmonic oscillator, the relaxation time is 50s.Find the time in which the amplitude and energy of the oscillator falls to1/e times its initial value.

3. An under damped harmonic oscillator has its amplitude reduced to 1/10th of its initial value after 100 oscillations. If its time period is 1.15s then calculate (i) damping constant (ii) relaxation time.

4. A quality factor of sonometer wire of frequency 240 Hz is 2000. Find the time in which the amplitude decreases to l/e2 of its initial value.

5. Describe how the amplitude of a weakly damped driven oscillator varies with frequency of the driving force. Discuss the cases when <<0.

Find the frequency for which amplitude becomes maximum, 0 is natural frequency of the oscillator.

6. A particle undergoing free vibration under a linear restoring force and a damping proportional to velocity is subjected to a periodic driving force of the form Fosint. Find an expression of the amplitude of the system at steady state.

7. If a weakly damped harmonic oscillator is driven by an external force, the amplitude of the steady state oscillations is 0.1 mm at very low values of ω and attains a maximum value of 10 cm when ω= ω0 =100 rad/s. Calculate (a) the Q factor (b) the time during which the energy decays to l/e of its initial value and (c) half width of the power resonance curve.

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8. Comment on the statement” The smaller the damping, the larger will be relaxation time and greater the quality factor”

9. Explain the phenomenon of resonance in the context of a damped oscillator undergoing forced vibration. Find an expression for energy at resonance.

10. Prove that the angular frequency at which maximum amplitude occurs differs slightly from the resonant frequency. Explain frequency response graph for different conditions of oscillations

11. A forced harmonic oscillator shows equal amplitude of oscillations at frequencies 1= 300 rad/s and 2 = 400 rad/s. Find the resonant frequency R at which the amplitude becomes maximum

12. Prove that when damping forces are present , the frequency of an oscillator is reduced by 12.5/Q2 where Q is the quality factor.

13. A mass 0.1 kg is suspended from a spring of constant 10 N/m. A resistive force Pv acts on the object, where p=0.1 Ns m-1 and v is the velocity. When an external force F = F0 sin ωt, with F0 = 1 N, is applied to the mass, what will be the amplitude in the steady at ω = 1 rad/s?

Tutorial Sheet :3 (Module I:Oscillations and Waves)

1. Given below is the equation of a progressive wave ,where time t is in second s and distance x in metres Y= a cos 240(t-x/312)

What are the values of the following quantities for this wave?(a)velocity of the wave ,(b)frequency of oscillation and (c)phase difference between two positions 0.50 meter apart .

2. Derive an expression giving the distribution of (i) velocity in a plane

progressive wave , (ii) acceleration in the same plane

progressive wave. 3. The vibrations of a string of length 60cm fixed at both ends are

represented by the equation y=4sin (xπ /15)cos(96πt) where x and y are in cm and t in sec. (i) what is the maximum displacement of a point at x=5cm(ii) where are the nodes located along the string (iii) what is the velocity of the particle at x=7.5 cm and t=0.25sec. (iv) write down the equations of the component waves whose superposition gives the above wave.

4. A S.H. wave of amplitude 8 cm traverses a line of particles in the direction of positive X-axis. At any given instant of time, for a particle at a distance of 10 cm from the origin, the displacement is cm and for a particle at a distance of 25 cm from the origin, the displacement is 4 cm. Calculate the wavelength.

5. A train of simple harmonic waves is traveling in a gas along the positive

direction of the x-axis with an amplitude equal to 2 cm ,velocity

300m/sec and frequency 400Hz.Calculate the displacement ,particle 3

velocity and particle acceleration at a distance of 4cm from the origin

after an interval of 5 seconds

6. Which method is suitable for the production of ultrasonic waves of the order of 10 Hz .Describe it.

7. A quartz crystal of thickness of 0.001 metre is vibrating at resonance. Calculate the fundamental frequency .Given Y for =7.9x10 N/m and ρ for quartz=2650kg/m .

8. Describe the piezoelectric method of producing ultrasonic wave’s .Give three names of piezoelectric transducers used in production of ultrasonic.

9. Explain in details how the ultrasonic pulse technique is used for non –destructive testing material.

10. A piezo electric X-cut crystal plate has a thickness of 1.6 mm. If the velocity of propagation of sound waves along the X-direction is 5760m/sec, calculate the fundamental frequency of the crystal.

11. Bats emit ultrasonic waves. The shortest wavelength emitted in air by a bat is about 0.33 cm.what is the highest frequency a bat can emit?

12. .A piezoelectric crystal slice has a thickness of 1mm.the velocity of sound in the crystal is 5700m/s.what will be the fundamental frequency in the crystal?

13. .Find the frequency of the first and second modes of vibration for a quartz crystal of piezoelectric oscillator .the velocity of longitudinal waves in quartz crystal is 5.5x10 m/s.Thickness of quartz crystal is 0.05m.

Tutorial Sheet : 4 (Module II:Wave Nature of light)

1. In the Young’s double slit experiment, interference fringes are formed using sodium light (with wavelengths 5890 Å and 5896 Å). Obtain the regions on the screen where the fringe pattern will disappear. Given the distance between slits d = 0.5 mm and distance between screen and the double slit plane D = 100 cm.

2. In a Fresnel’s biprism experiment the fringe width is observed to be 0.087 mm. What will it become if the slit to biprism distance is reduced to ¾th of the original distance?

3. In an experiment using sodium light of wavelength 5890 Å, an interference pattern was obtained in which 20 equally spaced fringes occupied 2.30 cm on the screen. On replacing sodium lamp with another monochromatic source of a different wavelength with no other changes, 30 fringes were found to occupy 2.80 cm on the screen. Calculate the wavelength of light from this source.

4. If one carries out the Young’s double slit experiment using microwaves of wavelength 1 cm, discuss the nature of the fringe pattern if d = 0.1 cm, 1 cm and 4 cm. You may take D = 100 cm. Is the equation for fringe width derived for the light waves will be applicable here?

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5. A radio station broadcasts simultaneously from two transmitting antennas at two different locations. Will your radio have a better reception with two transmitting antennas rather than one? Justify your answer.

6. In the Young’s double slit experiment, light waves coming from both slits have their phase shifted by half a wavelength. How would the fringe pattern change at the screen? How would the pattern change if the light coming from only one of the slits had its phase shifted by half a wavelength?

7. In the Young’s double slit experiment, the angle that locates the second dark fringe on either side of the central bright fringe is 5.4°. Find the ratio of the slit separation d to the wavelength of light.

8. Assume a plane wave incident normally on a plane containing two holes separated by a distance d. If we place a convex lens behind the slits, show that

the fringe width, as observed on the focal plane of the lens, will be , where f is

the focal length of the lens.

9. In the Young’s double slit experiment, consider two points on the screen, one corresponding to a path difference of 5000 Å and the other corresponding to a path difference of 40000 Å. Find the wavelengths (in the visible region) which correspond to constructive and destructive interference. What will be the colour of these points?

10. A soap film (n = 1.33) is 375 nm thick and is surrounded on both sides by air. Sunlight (wavelength range 380 nm to 750 nm) strikes the film nearly perpendicularly. For which wavelength(s) in this range does constructive interference cause the film to look bright in reflected light?

11. A mixture of red (λ = 661 nm) and green light (λ = 551 nm) shines perpendicularly on a soap film (n = 1.33) that has air on both sides. What is the minimum nonzero thickness of the film, so that destructive interference causes it to look red in reflected light?

12. A non-reflective coating of magnesium fluoride (n = 1.38) covers the glass (n = 1.52) of a camera lens. Assuming that the coating prevents reflection of yellow-green light (λ = 565 nm), determine the minimum non-zero thickness of the coating can have.

13. Two plane glass plates are placed on top of one another and on one side a paper is introduced to form a thin wedge of air. Assuming that a beam of wavelength 600 nm is incident normally, and that there are 100 interference fringes per cm, calculate the wedge angle.

14. In Newton’s ring experiment the diameters of 4th and 12th dark rings are 0.4 and 0.7 cm, respectively. Calculate the diameter of 20th dark ring.

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15. Newton’s rings are formed by monochromatic light between a flat glass plate and a plano-convex lens is viewed normally. Calculate the order of the dark ring which will have double the diameter of that of 40th dark ring.

16. Newton’s rings are formed by monochromatic light of λ = 640 nm. Assume that the point of contact to be perfect. Discuss the ring pattern as seen through the microscope if the lens is slowly raised vertically above the plate. Radius of the convex surface is 100 cm. What are the radii of the first and second dark rings when (a) the lens rests on the glass plate, (b) lens is raised through a height of λ/4, (c) lens is raised through a height of λ/2 ?

17.

In the above figure a plano-convex lens is resting on a flat glass plate. The thickness t = 1.37 X 10e-5 m. The lens is illuminated with monochromatic light (λ = 550 nm) and a series of concentric bright and dark fringes are formed. How many bright rings are there?

18. A convex surface of radius of radius 300 cm of a plano-convex lens rests on a concave surface of radius 400 cm and Newton’s rings are viewed with reflected light of wavelength 6000 Å. Calculate the diameter of the 10th bright ring.

19. In a Newton’s ring arrangement, light consisting of wavelengths and incidents normally on a plane convex lens of radius of curvature R resting on a glass plate. If the nth dark ring due to coincides with (n+1)th dark ring due to

, then show that the radius of the nth dark ring of is given by .

20. In a Newton’s ring arrangement if the incident light consists of two wavelengths 5000 Å and 5002 Å. Calculate the distance from the point of contact at which the rings will disappear. Assume that the radius of curvature of the curved surface is 400 nm.

Tutorial Sheet : 5 (Module II:Wave Nature of light)

1. Show that, for Fraunhofer diffraction at a single slit, the relative intensities of the successive maxima are approximately 1 : 4/92 : 4/252 : 4/492 ………..

2. A single slit is illuminated by light composed of two wavelengths 1 and 2. One observes that due to Fraunhofer diffraction the first minimum obtained for 1 coincides with the second diffraction minimum of 2. What is the relation between 1 and 2 ?

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t

3. Diffraction pattern of a single slit of width 0.5 cm is formed by a lens of focal 40 cm. Calculate the distance between the first dark and next bright fringe from the optic axis. ( = 4890 Å)

4. Light is incident normally on a grating 0.5 cm wide with 2500 lines. Find the angles of diffraction for the principal maxima of the two sodium lines in the

first order spectrum. Given that and .

5. In a diffraction phenomenon using double slit, Calculate (i) the distance between the central maximum and the first minimum of the fringe envelope and (ii) the distance between any two consecutive double slit dark fringes. Given data: Wavelength of light= 5000 Å, Slit width =0.02 mm, Spacing between two slits = 0.10 mm, Screen to slits distance = 100 cm.

6. Deduce the missing orders for a double slit Fraunhofer diffraction pattern, if the slit widths are 0.16 mm and they are 0.8 mm apart.

7. What is grating element? Show that only first order is possible if the width of the grating element is less than twice the wavelength of light.

8. A diffraction grating is just able to resolve two line of =5140 Å and =5140.85 Å in the first order. Will it resolve the line = 8037.20 Å and = 8037.50 Å in the second order ?

9. What is the ratio of resolving powers of two gratings having 15000 lines in 2 cm and 10,000 lines in 1 cm in first order? Each grating has lines in its 2.5 cm width.

10. A plane transmission grating has 15000 lines per inch. Find the angle of separation of the 5058 Å and 5016 Å lines of helium in the second order spectrum.

11. In the second order spectrum of a plane diffraction grating, a certain spectral line appears at an angle of 10°, while another line of wavelength 5 X 10 -9 cm higher appears at an angle 3’’ more. Find the wavelengths of the lines and the minimum grating width required to resolve them. (Given sin 10° = 0.1736 and cos 10° = 0.9848)

12. Consider a diffraction grating of width 5 cm with slits of width 0.0001 cm separated by a distance of 0.0002 cm. What is the corresponding grating element? How many diffraction orders would be observed for wavelength =5 X 10-3 Cm? Calculate the width of the principal maximum?

13. Light is incident normally on a grating of total ruled width 5 X 10 -3 m with 2500 lines in all. Calculate the angular separation of two sodium lines in the first order spectrum. Can they be seen distinctly?

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14. The wavelengths of sodium D lines are 589.6nm and 589nm. What is the minimum number of lines that a grating must have in order to resolve these lines in the first order spectrum?

Tutorial Sheet : 6 (Module II:Wave Nature of light)

1. The electric field components of a plane electromagnetic wave are, and . Draw the diagram showing

the state of polarization when (a) = 0, (b) = π/2, (c) = π/4.

2. A beam of vertically plane polarized light is normally incident on an ideal linear polarizer. Show that if its transmission axis makes an angle of 60° with the vertical, only 25% of the irradiance will be transmitted by the polarizer.

3. At what angle will the reflection of the sky coming of the surface of a pond (n = 1.33) completely vanish when seen through a Polaroid filter?

4. Light reflected from a glass plate (ng = 1.65) immersed in ethyl alcohol (ne = 1.36) is found to be completely linearly polarized. At what angle will the partially polarized beam be transmitted into the plate?

5. A right circularly polarized beam is incident on a calcite half-wave plate. Show that the emergent beam will be left-circularly polarized.

6.

A HWP (half wave plate) is introduced between two crossed Polaroids P1 and P2. The optic axis makes an angle 15° with the pass axis of P1 as shown in Figs (a) & (b). If an unpolarized light beam of intensity I0 in normally incident on P1 and if I1, I2 and I3 are the intensities after P1, HWP and P2 respectively, then calculate I1/I0, I2/I0, I3/I0.

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HWP

P1

P2

I0

I1

I2

I3

Z

X

Z

Y

15°

Pass axis of P1

7. A quartz plate cut perpendicular to the optic axis is to be used to rotate the plane polarized light through an angle of 90°. If wavelength of the light used is 5461 Å, find its thickness.

8. A beam of linearly polarized light is changed into circularly polarized light by passing it through a sliced crystal of thickness 0.005 cm. Calculate the difference in refractive indices of the two rays in the crystal assuming this to be of minimum thickness that will produce the effect. The wavelength of light

used is .

9. An unknown solution is suspected of containing particular substance of specific rotation Sλ = -51.4°. If a 15 cm length of this solution rotates sodium light (5893 Å) by 25.6°, what is the concentration of this particular substance?

10. A polarimeter tube is 30 cm long filled with a solution containing 15 gm of cane sugar per 100 cc is placed in the path of a plane polarized light. Find the angle of rotation of the plane of polarization. Specific rotation of sugar is 66.5°.

11. A length of 15 cm of a solution containing 40 gm of solute per liter causes a rotation of the plane of polarization of light by 60. Calculate the rotation of plane of polarization by a length of 22gm cm of a solution containing 100 gm of solute per liter.

12. Calculate the specific rotation if the plane of polarization is turned through 220, traversing 15 cm length of 20% sugar solution.

13. A calcite plate cut with its faces parallel to the optic axis is placed between two crossed nicols with its principal section at 350 with polarizer. Find a) the amplitudes, b) the internets of the O and E vibrations leaving the calcite. Find also (c) the relative amplitudes and (d) intensities leaving the analyzer.

14. A plate of thickness 0.020 mm is cut from calcite with optic axis parallel to the face. Given, μo = 1.648 and μe = 1.481 (ignoring variations with wavelength), find out those wavelengths in the range 4000 Å to 7800 Å for which the plate behaves as a half wave plate and also those for which the plate behaves as a quarter wave plate.

15. A beam of light is passed through a polarizer. If the polarizer is rotated with the beam as an axis, the intensity I of the emergent beam does not vary. What are the possible polarization states and how to ascertain the state of the light beam with an additional QWP?

16. A λ/4 plate is rotated between two crossed Polaroids. If an unpolarized beam is incident of the 1st Polaroid, discuss the variation of intensity of the emergent beam as the quarter wave plate in rotated. What will happen if have a λ/2 plate instead of a λ/4 plate

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Tutorial Sheet :7 (Module III :Electromagnetics)

1. Define the gradient of a scalar field. What is the physical significance of gradient?

2. Find the gradient of the functions (a) f(x,y,z) = x2 + y3 + z4 and (b) f(x,y,z) = exsin(y)ln(z).

3. Let r be the vector from some fixed point (x0, y0, z0) to the point (x,y,z) and r be its length. Show that (a)

(b) , where is a unit vector in the direction of r.

4. Explain the geometrical interpretations of the divergence and curl of a vector field.

5. Calculate the divergence of the following vector functions(a)

(b)

(c) 6. Show that for a scalar function ‘t’ and vector

function .7. Check the divergence theorem for the function .

Take volume the cube of side unity and situated at origin.8. Find the constant ‘a’ for which the vector

is solenoidal.9. Prove that

(a) where is a scalar function.(b) where A is a vector function.

10. Check the Stokes theorem for the square surface of side unity at origin. Given vector function and area vector .

11. Given a position vector , show that .

Tutorial Sheet : 8 (Module III :Electromagnetics)

1. What is electric flux? If electric field is given by , then

calculate the electric flux through a surface of area 400 units lying in y-z

plane.

2. Consider a uniform electric field E oriented in the X direction. Find the net electric flux through the surface of a cube of edge length l situated at origin.

3. State and prove Gauss’s law in electrostatics.

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4. Determine the electric intensity inside and outside a hollow spherical conductor carrying a charge q. Given R1 and R2 are the inside and outside radii.

5. Derive Coulomb’s law from Gauss’s law.6. Using Gauss theorem show that for points external to charged sphere, the

force exerted are the same as would be produced if all the charges were concentrated at the centre of the sphere.

7. A spherically symmetric charge distribution of radius R is characterized by the charge density function

for r R

= 0 for r > R.Calculate (a) total amount of charge, (b) electric field at internal and external points.

8. Use Gauss law to calculate the electric field intensity due to a uniformly charged sphere at external and internal points.

9. Find the electric field due to an infinite plane of positive charge with uniform surface charge density .

10. Explain why Gauss’s law can not be used to calculate the electric field near an electric dipole, a charged disc, or a triangle with a point charge at each corner.

Tutorial Sheet : 9 (Module III :Electromagnetics)

1. State and prove Ampere’s circuital law in electromagnetism.2. Write the Maxwell’s equations in both integral and differential form. Give

the physical significance of each equation.3. Derive Maxwell’s equations in differential and integral form.4. Show that equation of continuity is contained in Maxwell’s equations.5. Explain the propagation of plane electromagnetic waves in free space and

show that electric vector E and magnetic vector H are mutually perpendicular.

6. Define the Poynting vector. Derive Poynting theorem for flow of energy in an electromagnetic wave.

7. Calculate the magnitude of Poynting vector at the surface of the sun, if radius of sun = m and power radiated by the sun = watt.

8. What is intrinsic impedance? Show that the intrinsic impedance of the vacuum is 377 .

9. Outside the earth’s atmosphere the intensity of sunlight (solar constant) is 2 cal cm-2 min-1 (= ). Calculate (i) the value of the amplitudes of electric and magnetic fields (assuming light to be a plane wave).

(ii) the radiation pressure on the earth at normal incidence if it absorbs all the light. (iii) the radiation pressure on the earth if it acts like a perfect mirror.10. In a radio wave , calculate the magnitude of and Poynting

vector.11

11. The relative permittivity of distilled water is 81. Calculate refractive index and velocity.

12. If the magnitude of H in a plane wave is 1 amp/meter. Find the magnitude of E for a plane wave in free space.

13. A 60 watt point source radiates equally in all directions. Calculate the magnitudes of the peak value of electric and magnetic fields at a distance 10 m from the source.

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