PHY303 – Special Relativity (2015)






COURSE CODE:                    PHY303                                                                                                                                COURSE TITLE:                     Special Relativity                                                                                                        TIME:                                     3 Hours


INSTRUCTION: Answer question 1 and any three questions.



Speed of light ; mass of electron ;                        Electronic charge  ; Avogadro’s number                    Boltzmann constant ; Plank’s constant ;

1.(a)(i) What is time dilation in special relativity?                                                                        2 marks

(ii) Show how Lorentz transformation accounts for the contraction of bodies and retardation of clocks that are in motion.                                                                                                                         6 marks

(b)(i) Muons (mu mesons) are unstable particles with an average life span of  and speed of . They are created at attitudes of some thousands of kilometers in the atmosphere by cosmic rays incident upon the earth from outer space. With suitable calculations, explain why they are found on earth in profusion despite their short life span.  

           5 ½ marks

(ii) An observer in a rocket measures its length as  and orientation as  relative to the horizontal. Calculate the length and orientation of the rocket as it appears to a stationary observer on earth if the rocket’s speed is                                                                                                   4 marks

2.(a)(i) Explain what you understand by the term inertial reference frame of reference.     2 marks

(ii) Show that Newton’s second law of motion is invariant under Galilean transformation.  4 marks

(b)(i) A swimmer can swim with a speed  in the still water of a lake. In a stream in which the speed of the current is  (which, we assume, is less than ), the swimmer can also swim with a speed  relative to the water in the stream. Suppose the swimmer swims upstream a distance  which is equal to the width of the stream and then returns downstream to the starting point. Find the time taken to make the round trip and compare it with the round trip time taken to swim straight across the stream.                                                                                                                             4½ marks

(ii) The equation of an electromagnetic wave in free space is given as

Show that this equation is not invariant under Galilean transformation.                             7 marks

  1. (a)(i) In a Michelson-Morley experiment, an interferometer with arms of unequal length was used. One arm was parallel and the other perpendicular to the ‘ether wind’. Show that the number of fringes observed when the apparatus is rotated through  is

to the first order in                                                                                                             6 marks

(ii) Show that the Lorentz – Fitzgerald contraction can account for the null result of the Michelson-Morley experiment.                                                                                                                             3 marks

(b)(i) In a Michelson-Morley experiment, the lengths of the arms of the interferometer was found to be  and the wavelength of light used was . Calculate the total expected fringe shift when the apparatus is rotated through . Take the orbital speed of the earth to be .                                                                                 4 ½ marks

(ii) Briefly discuss two viewpoints that were suggested to retain the ether concept.          4 marks


4.(a)(i) Obtain the mass-energy relation . Show that in the limit as   that is at ordinary speeds, the relation reduces to                                                             7 marks

(ii) At what speed does the mass of a particle become 10 times its rest mass?                  3 ½ marks

(b) (i) Apart from the length contraction and time dilation, briefly discuss any two experimental evidence of the principles of special relativity.                                                                      4 marks

(ii) A spaceship moving away from the earth with velocity  fires a rocket whose velocity relative to the spaceship is  away from the earth. What will the velocity of the rocket be as observed from the earth?                                                                                                                 3 marks

  1. (a)(i) What is a four-vector and how is it different from a Euclidean vector? What do you understand by invariance of the space-time interval?           4 ½ marks

(ii) Show that the Lorentz coordinate transformation is an orthogonal transformation. 7 marks

(b)(i) In matrix notation, write down the components of the momentum and force four-vectors.

(ii) Show that the magnitude of the momentum four-vector is , where  is ,  the rest mass and  the speed of light.                                                                                       6 marks


6(a)(i) Describe the linear charge density as seen by two observers in different frames of reference and show that linear charge density is not Lorentz invariant.                                  3 marks

(ii) Show how magnetism arises given electrostatics and special relativity and obtain the relevant equations.                                                                                                                  6 ½ marks

b(i) Write down the components of the electric and magnetic fields  and  in the  frame in terms of the components  and  in the  frame, where both frames are inertial.  4 marks

(ii) Find the magnetic field of a point charge in uniform motion.                                  4 marks


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