Physics Tutioral Sheet 2014

Amity School of Engineering & Technology, NoidaApplied Physics I

Tutorial Sheet: 1 (Module I : Electromagnetic Theory)

1. If a vector , find the magnitude and the direction cosines of this vector.

2. If and , find (a) , and (b) .

3. If and , find the angle between and .

4. What do you mean by scalar and vector fields? Define operator.

5. Explain the terms (a) gradient, (b) divergence, (c) curl by bringing out their physical

significance.

6. Prove that: .

7. Prove that: (a) , (b) , (c)

.

8. If , find (a) , (b) ,

9. If is the position vector of a point, then show that .

10. The potential function in an electric field is represented as ,

where C is an arbitrary constant. Show that the electric field is radial.

11. If represents the electrostatic potential at a point, find the electric

field intensity at a point (3,2,-2).

12. Prove that the divergence of a vector field which obeys the inverse square law is zero.

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Tutorial Sheet: 2 (Module I: Electromagnetic Theory)

1. A vector field is given as , find (a) , (b) .

2. If is a scalar field and is a vector field, prove that

.

3. If , find the value of (a) , (b) at the point

(2,2,2).

4. Show that represents a conservative field.

5. Prove that: (a) , where is a unit vector, (b) , where is

the position vector.

6. If , calculate at the point (1,-2,-1).

7. Find , where is the distance of any point from the origin.

8. If is the position vector of a point, then find .

9. A vector function has the following components: , , and

, show that .

10. If is a position vector of a point, show that: (a) , (b) , (c)

, (d) .

11. Find the constant “a” for which the vector is

solenoidal.

12. State and prove (a) Gauss’s divergence and (b) Stokes’ theorem.

Tutorial Sheet: 3 (Module I: Electromagnetic Theory)

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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. State and prove Gauss’s law in electrostatics. Derive Coulomb’s law from Gauss’s

law.

3. State and prove Ampere’s circuital law in electromagnetism.

4. Write the Maxwell’s equations in free space in both integral and differential form.

Give the physical significance of each equation.

5. Derive Maxwell’s equations in differential and integral form.

6. Show that equation of continuity is contained in Maxwell’s equations.

7. Explain the propagation of plane electromagnetic waves in free space and show that

the electromagnetic waves propagate with the speed of light in free space. Also prove

that the em waves are transverse in nature.

8. Using Maxwell’s electromagnetic equations: and ,

show that (a) and (b) , where symbols have their usual meaning.

9. Using Maxwell’s electromagnetic equations: and , derive:

(a) Coulomb’s law in electrostatics, and (b) Equation of continuity.

10. Deduce an expression for the velocity of propagation of a plane electromagnetic wave

in a medium of dielectric constant Є and relative permeability µ.

11. If the amplitude of in a plane wave is 1 A/m, calculate the magnitude of for

plane wave in free space.

Tutorial Sheet: 4 (Module II: Wave Optics)

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1. (a)What are the conditions for observing interference fringes?

(b) What are coherent sources? How they are realized in practice? Explain why two

independent sources can never be coherent sources. Describe two methods for the

production of coherent sources.

2. 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.

3. In a Young’s double slit experiment, the angular width of a fringe formed on a distant

screen is 0.10. The wavelength of light used is 6000 Å. What is the spacing between

the slits.

4. Two coherent sources whose intensity ratio is 81:1 produce interference fringes.

Deduce the ratio of maximum intensity to minimum intensity in fringe system.

5. White light falls normally upon a soap film whose thickness is 5x10-5 cm and whose

index of refraction is 1.33. Which wavelength in the visible region will be reflected

most strongly?

6. 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.

7. 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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8. In a Newton’s ring experiment, the diameters of 5th and 25th rings are 0.3 cm and 0.8

cm respectively. Find the wavelength of light used. Take Radius of curved surface of

lens R = 100 cm.

9. Newton’s rings formed by monochromatic light between a flat glass plate and a

plano-convex lens are viewed normally. Calculate the order of the dark ring which

will have double the diameter of that of 40th dark ring.

10. 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 .


1. (a)Differentiate between interference and diffraction phenomena in light.

(b)Explain the difference between Fresnel and Fraunhofer type of diffraction.

2. 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 ………..

3. Light of wavelength 5000 Å is incident normally on a plane transmission grating of

width 3 cm and 15000 lines. Calculate the angle of diffraction in first order.

4. 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.

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5. 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.

6. 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?

7. The limits of visible spectrum are approximately 400nm and 700nm. Find the angular

width of the first order visible spectrum produced by a plane diffraction grating

having 15000 lines per inches when the light is incident normally on the grating.

8. 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.

9. How many orders will be visible if the wavelength of incident radiation is 4800 Å and

the number of lines on the grating is 25000 lines per inch.

10. 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?

11. 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?

12. (a) What do you understand by the term resolving power of a grating? Explain

Rayleigh criterion for the limit of resolution.

(b) Two plane diffraction gratings A and B have the same width of ruled surface but

A has greater number of lines than B. Which has greater intensity of fringes?

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1. Explain the following terms:

a. Plain of polarization

b. Optic axis of a crystal

c. Double refraction

d. Quarter and Half wave plates

e. Ordinary and Extra ordinary ray

2. Using two Nicol prisms, how would you find whether the given plate is a quarter wave

plate or a half wave plate or a simple glass plate?

2. If the plane of vibration of the incident beam makes an angle of with the optic

axis, compare the intensities of extraordinary and ordinary rays.

3. 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?

4. A right circularly polarized beam is incident on a calcite half-wave plate. Show that

the emergent beam will be left-circularly polarized.

5. Calculate the thickness of a quarter wave plate of quartz for sodium light of

wavelength 5893 Å. The refractive indices of quartz for E-ray and O-ray are equal to

1.5533 and 1.5442 respectively.

6. 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 .

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7. 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.

8. 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

quarter wave plate?

9. 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.

10. A 20 cm long tube containing 60 cc of sugar solution produces an optical rotation of

110 when placed in a polarimeter. Calculate the quantity of sugar contained in the

tube. The specific rotation of sugar is 660.

Tutorial Sheet: 7 (Module III: Lasers and Fiber Optics)

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1. What are the characteristics of laser beams? Describe its important applications.

2. Why two level lasers does not exist.

3. Calculate the number of photons emitted per second by 5 mW lasers assuming

that it emits light of wavelength 632.8 nm.

4. A certain ruby laser emits 1.00 J pulses of light whose wavelength is 6943 Å.

What is the minimum number of chromium ions in the ruby?

5. Explain (a) Atomic excitations (b) Transition process (c) Meta stable state and (d)

Optical pumping.

6. Find the intensity of laser beam of 15 mW power and having a diameter of 1.25

mm. Assume the intensity to be uniform across the beam.

7. Calculate the energy difference in eV between the energy levels of Ne-atoms of a

He-Ne laser, the transition between which results in the emission of a light of

wavelength 632.8nm.

8. Find the intensity of a laser beam of 10 mW power and having a diameter of 1.3

m. Assume the intensity to be uniform across the beam.

9. What is population inversion? How it is achieved in Ruby Laser. Describe the

construction of Ruby Laser.

10. Explain the operation of a gas Laser with essential components. How stimulated

emission takes place with exchange of energy between Helium and Neon atom?

11. What is the difference between the working principle of three level and four level

lasers. Give an example of each type. How a four level Laser is superior to a three

level Laser.

12. Explain why population inversion is essential for laser action to take place.

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Tutorial Sheet: 8 (Module III: Lasers and Fiber Optics)

1. What is total internal reflection and explain its importance for optical

communication.

2. Explain acceptance angle and acceptance cone of a fiber. What do you mean by

numerical aperture of a fiber? Derive expression for them.

3. The refractive indices of core and cladding materials of a step index fiber are 1.48

and 1.45, respectively. Calculate: (i) numerical aperture, (ii) acceptance angle, and

(iii) the critical angle at the core-cladding interface and (iv) fractional refractive

indices change.

4. Calculate the angle of acceptance of a given optical fiber, if the refractive indices

of the core and cladding are 1.563 and 1.498, respectively.

5. What is the carrier frequency for an optical communication system operating at

1.55 μm.

6. Determine the numerical aperture of a step index fiber when the core refractive

index n1 =1.5 and the cladding refractive index n2 =1.48. Find the maximum angle

for entrance of light if the fiber is placed in air.

7. An optical fiber has NA of 0.20 and a cladding refractive index of 1.39.

Determine the acceptance angle for the fiber in water which has refractive index

of 1.33.

8. Calculate the numerical aperture acceptance angle and critical angle of the fiber

from the following data: n1 =1.5 and n2 = 1.45.

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9. A step index fiber with a large core diameter compared with the wavelength of the

transmitted light has an acceptance angle in air of 220 and a relative index

difference of 3%. Determine

(1) Numerical aperture of the fiber

(2) The critical angle at the core-cladding interface

TUTORIAL SHEET: 9 (Module IV: Special Theory of Relativity)

1. What do you mean by Inertial and non Inertial frames of reference? Is earth an inertial

frame?

2. Describe the Michelson Morley experiment and discuss the importance of its

negative result.

3. Calculate the fringe shift in Michelson-Morley experiment. Given that: ,

, , and .

4. State the fundamental postulates of Einstein special theory of relativity and deduce

from them the Lorentz Transformation Equations.

5. What is proper length? Explain relativistic length contraction on the basis of special

theory of relativity?

6. Give an example to show that time dilation is real effect.

7. What do you mean by proper time interval? Explain relativistic time dilation on the

basis of special theory of relativity?

8. A rod has length 100 cm. When the rod is in a satellite moving with velocity 0.9 c

relative to the laboratory, what is the length of the rod as measured by an observer (i)

in the satellite, and (ii) in the laboratory?.

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9. How fast would a rocket ship have to go relative to an observer for its length to be

contracted to 99% of its length at rest?

10. A clock keeps correct time. With what speed should it be moved relative to an

observer so that it may appear to lose 4 minutes in 24 hours?

11. Prove that x2+y2+z2 = c2t2 is invariant under Lorentz transformation.

12. In the laboratory the ‘life time’ of a particle moving with speed 2.8x108m/s, is found

to be 2.5x10-7 sec. Calculate the proper life time of the particle.

13. At what speed should a clock be moved so that it may appear to loss 1 minute in each

hour?

14. Derive relativistic law of addition of velocities and prove that the velocity of light is

the same in all inertial frame irrespective of their relative speed.

15. Two particles come towards each other with speed 0.9c with respect to laboratory.

Calculate their relative speeds.

16. Rockets A and B are observed from the earth to be traveling with velocities 0.8c and

0.7 c along the same line in the same direction. What is the velocity of B as seen by

an observer on A?

17. Deduce an expression for the variation of mass with velocity. Also prove that no

material particle can have a velocity equal to or greater than the velocity of light.

18. A proton of rest mass is moving with a velocity of 0.9c. Calculate its

mass and momentum.

19. The speed of an electron is doubled from 0.2 c to 0.4 c. By what ratio does its

momentum increase?

20. A particle has kinetic energy 20 times its rest energy. Find the speed of the particle in

terms of ‘c’.

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21. State and prove the law of equivalence of mass and energy.

22. Prove the relation E2- p2c2 = m02c4, where p is the momentum.

23. At what speed does the kinetic energy of a particle equal to its rest energy?

24. What should be the speed of an electron so that its mass becomes equal to the mass of

proton? Given: mass of electron=9.1x10-31Kg and mass of Proton =1.67x10-27Kg.

25. An electron is moving with a speed 0.9c. Calculate (i) its total energy and (ii) the ratio

of Newtonian kinetic energy to relativistic energy. Given: and

.

26. (i) Derive a relativistic expression for kinetic energy of a particle in terms of

momentum. (ii) Show that the momentum of a particle of rest mass and kinetic

energy , is given by .

27. Calculate the mass and speed of 2MeV electron.

28. A particle of rest mass m0 moves with speed c/√2. Calculate its rest mass, momentum,

total energy and kinetic energy.

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Physics Tutioral Sheet 2014