UNIVERSITY OF SURREY SCHOOL OF PHYSICS AND CHEMISTRY DEPARTMENT OF PHYSICS

Transcription

1 Phys/Level3/3/5/Autumn Semester UNIVERSITY OF SURREY SCHOOL OF PHYSICS AND CHEMISTRY DEPARTMENT OF PHYSICS MPhys in Physics MPhys in Physics with Nuclear Astrophysics BSc Honours in Physics BSc Honours in Physics with Acoustics BSc Honours in Physics with Medical Physics BSc Honours in Physics with Management Studies BSc Honours in Physics with Nuclear Astrophysics LEVEL 3 PAPER 3 RELATIVITY Answer TWO questions Time allowed: 2 hours Internal Examiner: External Examiner: Dr S J Doran Professor P C Main Any pocket calculator without an alpha-numeric display may be used. The numbers at the end of each section of a question give an approximate indication of the marks available.

2 Phys/Level3/3/5/Autumn Semester (a) What, in 19th Century Physics, was meant by the luminiferous aether? 1 mark (b) (c) Explain what the Michelson interferometer was, how it worked and what it was designed to test. Why was the result of Michelson s experiment (and subsequent confirmations in collaboration with Morley) such a shock to the established ideas of the day? The 1887 Michelson-Morley experiment used a sodium lamp ( = 590 nm) and an interferometer with both arms of length 11 m. It could detect a shift of only fringes. If the aether existed, what is the maximum velocity with which the Earth could be moving through it and still be consistent with a null result? 10 marks The Figure shows schematically a stellar aberration experiment. The angle at which a star appears above the plane of the Earth s orbit is measured when the Earth is at point 1. It is related to the corresponding angle when the Earth is at point 2, 6 months later. Use the fact that ( /c, k) is a four-vector to show that cos 2r / c cos, 1 2r cos / c where is the angular velocity of the Earth in its orbit and r is the orbital radius. How was the difference in angles and erroneously interpreted by 18th Century physicists to imply the existence of the aether? (Only the principle is required; you need not try to reconcile the aether model with the formula above.) 9 marks y * Star x Side view z 2 x r * Plan view 1

3 Phys/Level3/3/5/Autumn Semester (a) (i) Explain what is meant by the form invariance of a physical law in Special Relativity. 1 mark (ii) Use the concept of form invariance and the two Maxwell equations E / 0 and B 0J 0 E t to deduce that J x transforms as a component of the four-vector (c, J). [You are not required to establish the transformations for J y, J z and.] 9 marks (b) Two electrons with the same x-coordinate both move parallel to the x-axis at speed u with respect to the lab reference frame. They are separated by a distance r. (i) (ii) Why is the repulsive force between the two electrons in the lab frame not what one would naïvely expect from Coulomb s Law? Compare the relevant components of the force four-vector, F = u)(f.u/c, f) in the lab frame and a frame moving with the electrons to derive an expression for this force in the lab frame. 5 marks What is the force between the two electrons as measured in a reference frame moving at speed v along the x-axis with respect to the lab reference frame? 5 marks [ You may assume the following results for two reference frames moving relative to each other at speed v along the x-axis: Ex Ex Bx Bx Ey ( Ey vbz ) By ( B v y 2 c Ez ) Ez ( Ez vby ) Bz ( B v z 2 c Ey ). ]

4 Phys/Level3/3/5/Autumn Semester (a) (i) Define the terms four-vector and proper time. By appropriately differentiating the (ct, r) prototype four-vector, derive the form of the velocity four-vector, U. What relativistic invariant is associated with U? 5 marks (ii) Einstein postulated the four-vector relation P = m 0 U and the equivalence of mass and energy. Using these hypotheses and appropriate definitions of m and p to show that, for any two inertial frames of reference S and S, E c p E c p 3 marks (b) The figure opposite shows a diagram of an experiment to test Special Relativity. Protons incident on a stationary target, T, give rise to neutral pions with a mean energy in the lab frame of 12 GeV. The pions, with rest-mass GeV and half-life s in their own rest frame, decay into two photons. Those travelling along AB are detected. D 1 measures the time at which the protons initially hit the target and another (moveable) detector, D 2, measures the arrival time of the decay photons. (i) Write down the Lorentz transformations for E and p given that the relative motion between S and S is with a velocity v along the x-axis. Hence, calculate the speed of the 0 in the lab frame. 4 marks (ii) Calculate the mean range of 0 in the lab frame of reference? (t mean = ln 2 t 1/2 ) 3 marks (iii) (iv) Explain why the decay path 0, i.e., producing just a single photon, is never observed. 3 marks As detector D 2 moves 30 m from A to B, the time difference defined by t t signal t from D signal from 2 D 1 changes. State the value of this change and explain briefly in terms of the rule for velocity addition why this experiment is important. 2 marks Figure for part (b) on opposite page

5 Phys/Level3/3/5/Autumn Semester Figure for question 3(b) Proton beam path T, D 1 D 2 A Shielding B

6 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 1 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 1B 8B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

7 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 2 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

8 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 3 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 2 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

9 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 4 of 4 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) 9 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

10 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 1 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 1B 9B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

11 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 2 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 5 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

12 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 3 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) 5 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

13 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 3 Page (please number each page) 1 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 5B 3B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

14 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 3 Page (please number each page) 2 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 4 3 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

15 EXAMINATION SOLUTION FORM, Spring 2000 Examiner (initials) SJD Level and Module 3REL Question Number 3 Page (please number each page) 3 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 3 2 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

16 Phys/Level3/3/4/Autumn Semester UNIVERSITY OF SURREY SCHOOL OF PHYSICS AND CHEMISTRY DEPARTMENT OF PHYSICS MPhys in Physics MPhys in Physics with Nuclear Astrophysics BSc Honours in Physics BSc Honours in Physics with Acoustics BSc Honours in Physics with Medical Physics BSc Honours in Physics with Management Studies BSc Honours in Physics with Nuclear Astrophysics LEVEL 3 PAPER 3 RELATIVITY Answer TWO questions Time allowed: 2 hours Internal Examiner: External Examiner: Dr S J Doran Professor J E Inglesfield Any pocket calculator without an alpha-numeric display may be used. The numbers at the end of each section of a question give an approximate indication of the marks available.

17 Phys/Level3/3/4/Autumn Semester (a) (i) What is meant by an invariant in Special Relativity? Show, using the Lorentz transformation, that ds 2 = c 2 dt 2 dx 2 is invariant for two reference frames moving at constant velocity with respect to each other along the x-axis. 4 marks (ii) (iii) What is a Minkowski diagram? Draw a Minkowski diagram with one spatial dimension. Show on it how spacetime can be divided into three regions based on the value of ds 2 and explain briefly the significance of these regions. 3 marks It is often stated, somewhat loosely, that faster-than-light travel would violate causality. Explain what is meant by this by considering the consequences of hypothetical faster-than-light signals in your diagram for (ii) above. Use a Minkowski diagram to illustrate how two spacetime events, at either end of a space-like trajectory, may have their order of occurrence reversed by a suitable choice of reference frame. 6 marks (b) Inertial reference frame S is moving at velocity v along the x-axis of the laboratory frame, S. In S, a straight rod, parallel to the x-axis and of proper length L 0, is moving in the y-direction at constant velocity u y and has no component of velocity along x. Show that, in S, the rod is inclined to the x-axis at an angle ( v) u 1 yv tan 2, c and find the length of the rod in S. 7 marks

18 Phys/Level3/3/4/Autumn Semester (a) (i) Explain the term standard configuration for two inertial reference frames S and S, as used in Special Relativity. For two such frames, write down the Lorentz transformation that connects space and time. 2 marks (ii) A particle moves at velocity u = (u x, u y, u z ) in frame S. Use the Lorentz transformation from (i) to derive an expression for u, the velocity measured in S, showing details of your working. 4 marks (b) Two spacecraft A and B, each of proper length 100 m, pass by the Earth with velocities u A = (0.8c, 0, 0) and u B = (0.6c, 0, 0) in the Earth s frame of reference. To an observer on the Earth, the rockets move apart with a separation velocity of 1.4c: true or false? Comment on your answer. 1 mark Calculate: (i) the velocity of B in the reference frame of A; 1 mark (ii) the length of B in the Earth s reference frame; 1 mark (iii) the length of B in the reference frame of A, showing that the result obtained by using the velocity from (i) is consistent with that obtained by using the triple- relation. 3 marks (c) The diagram below shows a simplified schematic form of a puzzling experiment performed by Fizeau in the late 1800 s. A beam of light propagates through a moving transparent fluid such as water. The velocity of the light beam through the fluid in the lab frame of reference is found to be c u v fluid, n i.e., the moving fluid appears to drag the light with it. Fizeau found that, to a good approximation, the drag factor was (1 1/n 2 ), where n is the refractive index of the medium. Using the velocity transformation for u x, as found in (a)(i) above, perform a Taylor expansion to first order. Assuming that v fluid is much smaller than the velocity of light in the fluid, explain why Fizeau got the result he did. (Hint: Remember that in a stationary fluid, the speed of light is c/n.) 8 marks Light source Moving liquid v fluid

19 Phys/Level3/3/4/Autumn Semester (a) (i) Define the term four-vector, including in your answer a vector equation containing the L 1 matrix, which describes the transformation of a four-vector between two frames of reference moving relative to each other with velocity v along the x-axis. 2 marks (ii) Show that the differential operator (/x 0, ) transforms like a four-vector. 6 marks (b) Starting from Einstein s first postulate of Special Relativity and the y-component of the Maxwell Equation B E, t show that the following electric and magnetic field components transform according to the equations E x Ex, Ez ( Ez vby ), By ( By E ) 2 z v c 8 marks (c) State the equation for the force acting on a particle of charge q and velocity v moving in the presence of applied E and B fields, as seen in frame S. In frame S a particle with charge +q moves with velocity v relative to, and in the plane of, a circular current-carrying loop of wire, as shown in the figure below. Draw diagrams showing the situation in S and S, the frame in which the particle is stationary. Show all relevant fields, forces and charges, and explain qualitatively where the force on the particle comes from in S. 4 marks I z y x v

20

21

22

23

24

25

26

27

28

29

30

31

32 Phys/Level3/?/?/Autumn Semester NB This is an old, pre-moderation version of the paper. Please check with the final version from the University. UNIVERSITY OF SURREY SCHOOL OF ELECTRONICS AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS MPhys in Physics BSc Honours in Physics LEVEL 3 PAPER? RELATIVITY Answer THREE questions Time allowed: 2 hours Internal Examiner: External Examiner: Dr S J Doran Professor J Inglesfield Any pocket calculator without an alpha-numeric display may be used. The numbers at the end of each section of a question give an approximate indication of the marks available.

33 Phys/Level3/?/?/Autumn Semester (a) (b) (c) (d) Rain falls vertically downwards at a speed of 10 ms 1. I have a small umbrella that just keeps me dry if I hold it vertically above me when I am standing still. At what angle to the vertical must I hold it if I am to stay dry whilst running for a bus at a speed of 4 ms 1? 3 marks Describe the principles of stellar aberration and explain how the above everyday problem is related to this phenomenon, assuming that light is composed of a stream of particles. 7 marks What had Bradley actually set out to find from the experiment in which he discovered stellar aberration? Given that the closest star is in fact about 4 light years away and that the radius of the Earth s orbit about the sun is approximately m, show quantitatively why he could not have measured this. 5 marks In the 19 th century, it was believed that light was not a stream of particles but a wave motion. Explain how the phenomenon of stellar aberration was interpreted in this case? What difficulties does the concept of aether, inherent in this interpretation, pose? 5 marks

34 Phys/Level3/?/?/Autumn Semester (a) Two inertial reference frames S and S are in standard configuration and their origins coincide at t = t = 0. The most general form of relationship between the spatial and temporal co-ordinates is a linear one, viz. x Ax Bt x Cx Dt, where A, B, C and D are constants that are not independent of each other. (i) (ii) Explain the terms inertial and standard configuration. Given the definitions of S and S, why must the above equations be linear? 2 marks By considering the motion of the origin of S in S and of S in S, eliminate B and D from the equations for x and x. State Einstein s second postulate of relativity (concerning the speed of light) and apply it in both frames to a light ray leaving the origins at t = t = 0. Hence eliminate C from your equations. 4 marks (iii) By considering what would happen in the situation where S and S are swapped and in which the sign of v is reversed, show that x ( x vt) x ( x vt) and define. Hence, or otherwise, obtain equations explicitly defining t in terms of t and x. 6 marks (b) A rod is moving parallel to the x-axis of frame S at speed v. At a given instant, t 0 in S, the rest frame of the rod, the ends A and B of the rod correspond to spacetime events (x A, t 0 ) and (x B, t 0 ). In the lab frame S, the corresponding events (x A, t A ) and (x B, t B ) occur at different times, such that x A = 2a, x B = a, t A = a/2c, t B = a/c. Calculate: (i) the velocity v of the rod, as seen in S; 3 marks (ii) the time t 0 ; 2 marks (iii) the proper length of the rod; (iv) the length of the rod as seen in S. 2 marks 1 mark

35 Phys/Level3/?/?/Autumn Semester (a) (b) (c) Explain what is meant by the term time dilation. Starting from the Lorentz equations, derive a relation between a proper time interval, measured in frame S and the corresponding interval t in another reference frame S in standard configuration with S. 2 marks Describe the classic muon decay experiment that confirmed the predictions of Special Relativity concerning time dilation. 8 marks For two frames S and S in standard configuration, write the Lorentz equations connecting energy and momentum as a four-vector equation, with the appropriate L 1 matrix written out in full. Define any symbols you introduce. 3 marks (d) A particular galaxy has diameter 10 5 light years. It is traversed by a high-energy cosmic ray proton (E = ev in the rest frame of the galaxy). Calculate: (i) the time taken for the particle to cross the galaxy, as seen in the rest frame of the galaxy; 3 marks (ii) the time taken in the proton s rest frame; 2 marks (iii) the diameter of the galaxy in the proton s rest frame. 2 marks

36 Phys/Level3/?/?/Autumn Semester (a) (i) Starting from the definition of the velocity four-vector, U = (u)(c, u) and making use of the Einstein mass-energy relation, derive an expression for the energy-momentum four-vector, P. 3 marks (ii) By further defining the four-force F as dp/d, where is the proper time, show that 1 de F ( u), ( u) f. c dt Assuming that any changes in the particle s energy are changes in kinetic energy as a result of an applied force (rather than changes in internal energy caused by heating), show that F may also be written 1 F ( u) f. u, f. c 7 marks (e) A particle is travelling with speed u 0 along the x-direction. At time t = 0, when the particle is at the origin, a constant force f y starts to act on the particle in the y- direction. (i) If u 0 <<c, show that the path of the particle for t > 0 is given by r( t) u t i 0 f y m 2 0 t 2 j 2 marks (ii) Obtain the corresponding formula for the case where u 0 is a significant fraction of the speed of light and sketch both trajectories on the same diagram. 3 marks (iii) Show that the force on the particle in its rest frame is directed along the y direction at t = 0 and has magnitude (u 0 )f y, but that at subsequent times, the force in the particle s rest frame is no longer parallel to y. 5 marks

37

38

39

40

41

42

43

44

45

46

47

48

49 Phys/3/3/6/03 1 UNIVERSITY OF SURREY c SCHOOL OF ELECTRONICS AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Programmes in Physics LEVEL HE3 PAPER 3 RELATIVITY Time allowed: 2 hours Attempt THREE questions Internal Examiner: Dr S J Doran External Examiner: Professor J E Inglesfield The only University approved calculators are Casio models FX115MS, FX115W or FX115S for September 1998 entry onwards. The numbers at the end of each section of a question give an approximate indication of the marks available. SEE NEXT PAGE...

50 Phys/3/3/6/03 2 Question 1 (a) (i) Write down the Lorentz equations for converting space and time co-ordinates from an inertial frame S to another, S, moving at constant velocity v relative to S along the x-axis. Define any quantities you introduce. 3 marks (ii) By expressing the equations from (i) in differential form, or otherwise, derive the velocity transformation equations for u x and u y. 4 marks (b) An alternative notation for the Lorentz equations is ( ) ( x cosh φ sinh φ ct = sinh φ cosh φ )( x ct ). Show that this is consistent with (a)(i) when tanh φ = v/c, and that the sinh and cosh formalism incorporates within it the correct definition of γ. [Note: sinh φ = 1 2 (eφ e φ ); cosh φ = 1 2 (eφ + e φ ); cosh 2 φ sinh 2 φ =1.] 5 marks (c) A missile is fired along the x-axis with velocity u relative to reference frame S. A rapid spacecraft matches velocities with this missile and fires a second missile at a velocity u relative to itself. (i) Find the velocity u 2 of the second missile as seen in S. Express your answer in terms of u and also in terms of φ, where tanh φ = u/c. [Note: tanh(α + β) = (tanh α + tanh β)/(1 + tanh α tanh β).] 4 marks (ii) Given a suitable spacecraft, the procedure of matching velocities with the missile and firing another missile could continue indefinitely. Show that, if this was done n times, the final missile would be travelling at velocity u n with respect to S, where u n = c ( e 2nφ ) 1. e 2nφ +1 4 marks SEE NEXT PAGE...

51 Phys/3/3/6/03 3 Question 2 (a) Consider the four-vectors A and B. Write down the scalar (dot) product of A and B (i) in terms of the individual components A µ and B µ (µ =0,1,2,3) (ii) as a matrix equation involving a metric. 4 marks (b) What does the term invariant mean in Special Relativity? Show that the scalar product of any two four-vectors is an invariant. 6 marks (c) (i) Draw a Minkowski diagram with one spatial dimension to illustrate a space-like, a time-like and a light-like interval. What do we know about s 2 in each of these cases? 4 marks (ii) Draw a second Minkowski diagram to illustrate a base reference frame S and two further frames of reference S 1 ad S 2 moving at speeds v 1 > 0 and v 2 < 0 with respect to S. Why is it difficult to compare distances easily between these reference frames on your diagram? Add to your diagram the locus of all points P for which the squared space-time distance s 2 between O and P is equal to the constant a 2. How does this curve help us calibrate the axes? 6 marks SEE NEXT PAGE...

52 Phys/3/3/6/03 4 Question 3 (a) (i) Explain the concept of a four-vector, including in your answer an appropriate matrix equation. 4 marks (ii) Write down an expression for the energy-momentum four-vector P in terms of E, c and p. 1 mark (b) Compton scattering is the process by which a photon collides with a free electron. In the diagram, P 1 and P 2 represent the energy-momentum four-vectors before collision of the photon and electron respectively, whilst P 1 and P 2 are the corresponding quantities afterwards. (i) Write down an equation expressing the consevation of four-momentum and show that it leads to P P P (P 1.P 2 P 1.P 1 P 2.P 1) = P marks (ii) Explain why P 2 1 = P 2 1 =0in the above equation and show that P 2 2 = P 2 2 = m 2 ec 2 by looking at the electron in its own rest frame. 4 marks (iii) Show that the cross terms may be written as P 1.P 2 = m e E; P 1.P 1 = EE (1 cos θ); P c 2 2.P 1 = m e E. [You may find it helpful to make a choice of axes such that x lies horizontally along the page, y is vertical and z is out of the page towards you.] 4 marks SEE NEXT PAGE...

53 Phys/3/3/6/03 5 (iv) Hence, or otherwise, show that 1 E 1 E = 1 cosθ m e c 2 and interpret this result in terms of the photon s wavelength. 4 marks SEE NEXT PAGE...

54 Phys/3/3/6/03 6 Question 4 (a) Explain, in general terms and without mathematical detail, how Special Relativity views the origin of the Lorentz force f = q(e + v B) and how it is interpreted in inertial reference frames moving relative to each other. 8 marks (b) Consider a stationary line charge with charge per unit length ρ in frame S. Show how Lorentz contraction of this charge in moving reference frame S leads to the transformation equations ρ = γρ and j x = γvρ. Demonstrate that these equations are consistent with the statement that J =(cρ, j) is a four-vector. 6 marks (c) An infinite straight wire is arranged parallel to the x-axis. In its proper reference frame S, the wire is electrically neutral and a current I x flows along it. A point charge q moves at velocity u parallel to the wire and at a distance r from it. Consider the force felt by the charge in its own rest frame. Is this force electrostatic or magnetic in origin? Justify your answer in terms of the charge distribution on the wire in S 6 marks FINAL PAGE

55 EXAMINATION SOLUTION FORM, Autumn 2003 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 1 of 2 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 3B 3B 5 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

56 EXAMINATION SOLUTION FORM, Autumn 2003 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 2 of 2 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) 4 4 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

57 EXAMINATION SOLUTION FORM, Autumn 2003 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 1 of 2 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) 4B 6B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

58 EXAMINATION SOLUTION FORM, Autumn 2003 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 2 of 2 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) 4B s 2 > 0 s 2 = 0 s 2 < 0 6 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

59 EXAMINATION SOLUTION FORM, Autumn 2003 Examiner (initials) SJD Level and Module 3REL Question Number 4 Page (please number each page) 1 of 2 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) 8B 6 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

60 EXAMINATION SOLUTION FORM, Autumn 2003 Examiner (initials) SJD Level and Module 3REL Question Number 4 Page (please number each page) 2 of 2 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) 6 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

61 Phys/LEVEL HE3/3/6/Autumn Semester UNIVERSITY OF SURREY c SCHOOL OF ELECTRONICS AND PHYSICAL SCIENCES DEPARTMENT OF PHYSICS BSc and MPhys Programmes in Physics LEVEL HE3 PAPER 3 RELATIVITY Time allowed: 2 hours Attempt THREE questions Internal Examiner(s): External Examiner: Dr S J Doran Prof J E Inglesfield The only University approved calculators are Casio models FX115MS, FX115W or FX115S for September 1998 entry onwards. The numbers at the end of each section of a question give an approximate indication of the marks available. SEE NEXT PAGE...

62 Phys/LEVEL HE3/3/6/Autumn Semester Question 1 E = mc 2 and F = ma are two of the most famous equations in Physics. This question will examine the interpretation in Special Relativity of various terms in these equations. (a) (i) Explain what the constant c is and its significance in Special Relativity, quoting Einstein s Second Postulate. 3 marks (ii) Mass and energy are equivalent according to the above formula. Explain quantitatively, defining any terms you introduce, what happens to a particle s mass as it is accelerated from rest. Show how this leads to the idea that objects with a non-zero rest mass can never travel at the speed of light. 3 marks (iii) Derive an expression for the kinetic energy of a relativistic particle and show, by binomial expansion, that, at low speeds, this is the same as the classical formula KE = 1 2 mv2. 3 marks (b) (i) Starting from the four-vector equation F = dp dτ, define P and τ and show that the four-force F can be written F = γ(u)(f.u/c, f), where u is the velocity of a particle and f is the three-force acting on it. 5 marks (ii) A particle of rest mass m 0 is travelling along the x-axis at speed u, as shown in the figure below. It experiences a force at 45 to the positive x-axis as measured in a stationary observer s frame of reference. Show that the direction of the particle s acceleration is not parallel to the applied force, but at an angle of θ =tan 1 [γ(u) 2 ] to the positive x-axis. 6 marks f 45 u SEE NEXT PAGE...

63 Phys/LEVEL HE3/3/6/Autumn Semester Question 2 Write brief notes on the following topics: (a) time dilation, (b) length contraction. In each case outline the nature of the phenomenon, detailing the relevant relativistic equations and giving the experimental evidence supporting the idea. 20 marks SEE NEXT PAGE...

64 Phys/LEVEL HE3/3/6/Autumn Semester Question 3 (a) (i) Two reference frames S and S are in standard configuration. Explain what this means and write down a set of equations relating the (t, x, y, z) coordinates in S to the corresponding (t,x,y,z ) in S. Define any symbols you use. 4 marks (ii) Rewrite the equations from (i) in terms of a single four-vector equation involving the operator 1. Again, define any symbols introduced. 4 marks (b) A particle observed in frame S has energy 5 GeV and momentum 3 GeV/c. (i) What is its speed (as a fraction of c)? (ii) What is its rest mass (in GeV/c 2 )? 3 marks 3 marks (iii) A frame S is moving at velocity v relative to S, co-linearly with the particle. As seen in S, the momentum of the particle is 4 GeV/c. What is the particle s energy in S? 2 marks (iv) Calculate the velocity of S with respect to S. 4 marks SEE NEXT PAGE...

65 Phys/LEVEL HE3/3/6/Autumn Semester Question 4 (a) (i) State Einstein s first postulate of Special Relativity and explain how this leads to the concept of form invariance in physical equations. Illustrate your idea using Maxwell s Equations of electromagnetism, explaining without mathematical detail the meaning of any symbols you introduce. 3 marks (ii) Starting from the Lorentz Equations and noting that A u = φ A u φ + ψ u for any A = A[ φ(u, v), ψ(u, v) ], where u, v, φ and ψ are general variables, show that ( x = γ x v ) c 2 t and ( t = γ t v ). x A ψ 5 marks (iii) Hence, derive an expression for the y-component of curla (i.e., as measured in S) in terms of the operators in S (i.e., / x, etc.), where A is a general vector. 4 marks (b) The following equations relate the electric and magnetic fields in two reference frames in standard configuration: E x = E x ; E y = γ(e y + vb z); E z = γ(e z vb y) B x = B x ; B y = γ(b y ve z /c2 ); B z = γ(b z + ve y /c2 ) (i) Using your results from (a), show that if Maxwell s Laws hold in S, then the Maxwell Equation in S E = B t is satisfied by the y-component of B. 8 marks FINAL PAGE

66 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 1 of 4 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 3B 3B 3B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

67 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 2 of 4 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 5B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

68 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 3 of 4 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 2 4 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

69 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 1 Page (please number each page) 4 of 4 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

70 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 1 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 10B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

71 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 2 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 10B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

72 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 2 Page (please number each page) 3 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not MARKS (to total 20 for each question) Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

73 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 3 Page (please number each page) 1 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 4B 4B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

74 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 3 Page (please number each page) 2 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

75 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 3 Page (please number each page) 3 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 4 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

76 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 4 Page (please number each page) 1 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 3B 5B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

77 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 4 Page (please number each page) 2 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 4B Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)

78 EXAMINATION SOLUTION FORM, Spring 2004 Examiner (initials) SJD Level and Module 3REL Question Number 4 Page (please number each page) 3 of 3 Use BLACK ink only Writing must be LEGIBLE Indicate whether bookwork or not B = Bookwork MARKS (to total 20 for each question) 8 Please keep written material within the box to make photocopying easier For cut and paste, the rectangle immediately above is 8.1 h 6.5 w (20.5cm 16.5cm)