MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 8

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics October 31, 003 Due: Friday, November 7 (by 4:30 pm) Reading: Chapter 8 in French. Problem Set 8 Reminder: Quiz will be given on Friday, November 14, in the lecture hall beginning promptly at :00 pm. It will be designed as a 1 hour exam, but if no other class has claims on the room after 3:00, we may allow people to work for an additional 5 minutes or so. The quiz will cover the material since Quiz 1, including relativistic dynamics and the transformation of Coulomb s law; but not the subject we are just starting on Newtonian Cosmology. Problem 1 E p c is a Lorentz Invariant Start with the Lorentz transformations for energy and momentum, and show explicitly that E p c = E p c. Problem Relativistic Elastic Collision A relativistic particle of rest mass M 0 moves along the x axis of S with motion characterized by β 0 and γ 0. It collides with an identical particle that is at rest. After the collision it is observed that both particles are moving at angles ±θ with respect to the x axis. a. Show that the velocity of the CM frame (S frame) with respect to S is given by ( ) 1/ γ β CM = γ 0 β 0 /(γ 0 + 1) γ CM = [Hint: one way to proceed is to use the relation: β CM = p TOT c/e TOT.] b. Argue that before the collision, both particles have the same energy and magnitude of momentum in the CM frame, and that these quantities are characterized by γ CM and β CM. c. Argue that after the collision both particles are moving at right angles to the x axis in the CM frame. d. For each particle, after the collision, find p x and p y back in the S frame; express your answers in terms of γ CM and β CM. 1

2 e. Compute tan(θ) = p y /p x and show that tan(θ) = 1/γ CM = [/(γ 0 + 1)] 1/. f. In lecture we worked this same problem by directly conserving energy and momentum. There we found cos(θ) = [(γ 0 + 1)/(γ 0 + 3)] 1/. Show from simple trigonometry that this result is equivalent to that found in part (e). Problem 3 Pair Creation Two protons of mass 1 GeV collide to produce a particle of rest mass 300 GeV. The two protons remain after the collision. Find the threshold energy for this particle production to occur if (i) one of the protons is initially at rest, and (ii) if both protons have the same energy and collide head on. Comment on the relative efficiency of colliding beam accelerators vs. those using fixed targets. Problem 4 Proton Electron Photon System In the S inertial reference frame a proton of rest mass M p,0 is at rest at the origin. A photon with momentum p ν = M p,0 c moves in the y direction. An electron with momentum p e = M p,0 c moves in the x direction. (Note the photon and electron are traveling in orthogonal directions in S.) To make the calculations a little simpler, take the electron energy to be M p,0 c, and justify this as a reasonable approximation. a. Find the magnitude and direction of the total momentum of this system. b. What is the magnitude and direction of the transformation velocity needed to move into the zero momentum frame, S? c. Find the energy in the CM (S ) frame. Problem 5 Relativistic Transformation of Acceleration Starting from the definition of acceleration, a = du/dt, derive the x and y components of acceleration in an arbitrary inertial frame S moving with speed v along the x axis of S. The answers are: a x = a y = a x γ 3 (1 + vu x /c ) 3 a y γ (1 + vu x /c ) where the expression for a y is valid only for the case where either u y = 0 or a x = 0. In general, the transformation of a y is rather messy.

3 Problem 6 Constant Acceleration French, problem #7 8, Chapter 7, page 6. [Hint: You should end up integrating γ 3 du/dt = g to find u(t); then integrate udt = dx to find x(t). You can either look up the first of these integrals, or do it yourself with the substitution u = c sin(φ).] Problem 7 Ion Thruster Rocket A rocket with an ion thrust engine ejects xenon atoms (of atomic weight = 130) with a kinetic energy of 5 kev per atom. The rocket consists of 99.9% xenon fuel and only 0.1% structure. (See also Optional Problem B.) a. Derive the rocket equation: du 1 dm 0 V ex 1 dm 0 V ex = = dt γ 3 dt M 0 γ dt M 0 b. What is the exhaust velocity, V ex, of the xenon atoms in units of km s 1? c. If the rocket starts at rest in frame S, find its speed,u, in S after it has expended its fuel (in units of km s 1 ). Problem 8 Transforming Coulomb s Law 1 French, problem #8 1, Chapter 8, page 67. Problem 9 Transforming Coulomb s Law At t = 0 in S, charge q 1 (the source charge ) passes the origin with speed v in the x direction. At the same time, charge q is located at {x, y, t} = {0, y, 0}, and moving with speed u x in the x direction. Find the force on charge due to charge 1. Identify the electric and magnetic components of the force. Problem 10 Retarded Positions This problem is modeled after French, problem #8, Chapter 8, page 67. A charge q moves along the x axis with constant speed β. At time t = 0 the charge is located at x = 0; at an earlier time, t, the charge was at x = βct. Consider an arbitrary point P at {x, y}. Find the time, t 0, after which, no information from the particle can be communicated to the point P until after time t = 0. 3

4 Problem 11 Transformation of Electric and Magnetic Fields As you will learn in 8.07 (see also Optional Problem A), the general transformation for electric fields is: E x = E x Ey = γ(ey + βb z ) E βby z = γ(e z ) a. Consider the setup in Figure 8 of French (page 38). Explain why his equation F x = F x (on page 39) is consistent with the first of the transformations for the E field listed above. b. Consider the setup in Figure 8 3 of French (page 40). Explain why his equation F y = γf y (on page 41) is consistent with the second of the transformations for the E field listed above. The following problems are for discussion in recitation sections Problem A General Transformation of E and B Fields Start with Faraday s Law: 1 B E =, c t and select one of the vector components, e.g., E y E x = 1 B z. x y c t Now, convert the partial derivatives { / x, / y, / t} to the corresponding partials in S, { / x, / y, / t }, using the techniques established in Problem Set #3. In keeping with the postulates of Special Relativity, require that Faraday s law take the same form in S. E y E x 1 B z =, x y c t Equate terms in your transformed equation to the corresponding ones in the above equation to show that x y z E = E x E = γ(e y βb z ) B = γ(b z βe y ) The remaining three transformations [Ez = γ(e z + βb y ), B y = γ(b y + βe z ), B x = B x ] can be found from the other vector components of the Faraday equation. 4

5 Problem B Relativistic Rocket Problem (a) Show that in the rest frame of the rocket, for non relativistic dynamics: dm 0 du ΔM 0 V ex = M 0 Δu or V ex = M 0. dt dt where M 0 is the rest mass of the rocket, ΔM 0 is the differential mass loss in the exhaust, V ex is the speed of the rocket exhaust (in the rocket s frame, S ), and Δu is the change in the rocket s speed after ejecting ΔM 0. This is correct unless V ex is comparable with the speed of light. In a later part of the problem we look at the case of relativistic ejecta. (b) Consider first non relativistic motion of the rocket in a fixed (stationary frame). Note that in the non relativistic case a = a and F = F. Thus, the rocket equation in S is: dm 0 du V ex = M 0. dt dt If the rate of mass loss in the exhaust is known as a function of time, then the above expression can be directly integrated to find u(t) since M 0 (t) is known. Whether or not the rate of exhaust ejection is known or not, the dt s can be canceled on both sides and the equation integrated to find u as a function of the initial and final masses of the rocket. Show that this leads to: ( ) M initial u = V ex ln. M final (c) Suppose that the rocket in part (b) is firing its engine vertically in a gravitational field. Further assume that the atmospheric drag is negligible (not the case in the Earth s atmosphere, but OK on the Moon), and that the rocket motor is on while the rocket covers a distance that is relatively short compared with the radius of the gravitating body such that gravity, g, can be taken to be approximately constant. Show that if the rocket starts from rest, and ejects its exhaust uniformly in time (i.e., dm 0 /dt = constant = Ṁ 0 and M 0 = M 0,initial Ṁ 0 t), the rocket equation becomes: du dt V ex M = ( ) g. M initial M t Show that if the rocket is fired for a total time T, the above equation is easily integrated to yield u(t): [ ] [ ] u(t ) = V ex ln M initial M initial Ṁ T gt = V ex ln M initial M final gt (d) Next, consider the case where the rocket motion is at least somewhat relativistic, but the exhaust velocity is not. Then, the equation in part (a) tells us that in S the rocket acceleration is: a = du = V ex dm 0. dt M 0 dt 5

6 Next, transform the acceleration to the S frame to find: du 1 V ex dm 0 a = =. dt γ 3 M 0 dt From time dilation show that: du 1 V ex dm 0 =. dt γ M 0 dt Finally, cancel the dt s and integrate to find: ( ) V /c M i u M f 1 = ( ) c V /c M i + 1 M f, where V, M i, and M f are shorthand for V ex, M initial, and M final, respectively. (e) Relating the answers from parts (b) and (d): Show that for exhaust speeds much lower than c, the answer from part (d) matches the non relativistic expression in part (b). Show that unless V ex is a substantial fraction of the speed of light, the rocket will never attain truly relativistic speeds. (f) Relativistic fuel ejection: First, find the relativistically correct momentum of the exhaust in the rocket s frame. Assume that a fraction f of the rest mass of the fuel is converted to energy, and that all of this energy goes into the kinetic energy of the ejected fuel that has not been converted to energy. So, an energy equal to f ΔM 0 c goes into the kinetic energy of the mass (1 f )ΔM 0 c. Show that γ and β of the exhaust equal: 1 γ ex = 1 f β ex = f f. Finally, show that the momentum of the exhaust is: Δp = f f ΔM 0 c. Thus, for the relativistic rocket with a relativistic exhaust speed, show that the speed of the rocket is just that given in part (d) but with V ex replaced by f f c. Show that as f 1, V ex c, and that as f 0, V ex f c. 6