An electron and a proton are separately accelerated by accelerators that point a

Question

An electron and a proton are separately accelerated by accelerators that point at each other. The particles then approach each other as shown below. In each case the particle is accelerated from rest and an amount of work equal to 10mc^2 is done, where m is the rest mass ("the mass") of the particle. a) Show (turn off the calculator) using relativity that each particle will emerge from its respective accelerator with exactly the same speed as measured by an observer at rest with both accelerators. Determine that speed as a multiple (fraction) of c. b) Which particle has gained more energy through the acceleration process or is this also the same for the two particles? Explain how you know. c) Imagine that the accelerators are linear (i.e. straight evacuated tubes) with electric fields, directed so as to do positive work on the particle in question, that are uniform and of equal magnitude. Which accelerator is longer? How do you know? Find the ratio of the lengths of the accelerators. These are proper lengths as measured in the lab frame. A muon created via a high energy collision in the atmosphere travels 1954 m in 6.6 times 10^-6 s before decaying into three other particles: an electron and two neutrinos This is observed by scientists using instruments on a research balloons. a) Muons have a rest mass of approximately 2.0 times 10^-28 kg. Use this number to estimate (using relativity please) the amount of energy carried away by the three other particles (total) as measured by the detector in one of the balloons. Note that this muon is not at rest relative to the balloon when it decays and when it decays it is gone (poof!) replaced by the three other particles. b) How much energy would the particles carry away if the decay were observed from a reference frame at rest with the muon? c) What type of energy (e.g. kinetic, potential, thermal) does the balloon observer measure that the observer in part b does not? Explain your reasoning. a) Determine the velocity of the electron in question 1 relative to the proton as the particles emerge from the accelerators. Get the setup right: what's u, what's v etc.. b) How is your result from part a consistent with the "velocity of light" postulate of special relativity (see text and/or its consequences? c) Is the energy of the electron the same in the proton frame as it is in the lab frame? You have to use relativity. Show how you know.

Explanation / Answer

Ans:- a particle accelerator from rest & amount of work done 10 mc2 & the mass is rest of particle for given condition.

We know relativistic energy is related to Einstein relationship for energy E = mc2  

So the relativistic energy of a rest particle can be expressed in momentum. also the relativistic energy expression rest mass energy & the KE of motion. so the speed c

KE = mc2 - m0c2

So the KE of a particle as the Speed are same of the particle energy of rest mass energy. So if the

Electron rest mass = 8.2 x10-14 J = 0.511 MeV/c2

Proton rest mass = 1.673 x10-27Kg = 938.3 MeV/c2

E = hf       (Planck constant , fraction of the rest mass)

   = 6.626 x10-34 X 0.001   J.s

   = 6.6 X 10-37   J.s

E = mc2

6.6 X 10-37 =8.2 x10-14 c2

c = square root ( 6.6 X 10-37 / 8.2 x10-14 )

c = 8.97 x 10-15 s

E = mc2

6.6 X 10-37 =1.673 x10-27 c2

c = square root ( 6.6 X 10-37 / 1.673 x10-27)

c= 1.986 x 10-5 s

(b). Electron particle gain more energy through the acceleration process. we know electron size is very vast as compare to the Proton .so requirement more energy to move around the orbit .

(c) The accelerator are liner in vacuum tube in electrical field. the photon accelerator is longer . we electron the +ve charge practical. but photon is neutral charge practical in electrical field. Photon is does not effect the accelerator. so the photon accelerator is longer as compare to electron accelerator. ratio of length of the accelerator depend on the photon wavelength. so using the de Broglie wavelength

= h/mv Suppose rest energy photon = 0.5 MeV

Photon KE= 15 keV , rest mass = m. So we need proper wavelength to measure in the lab frame.

= h / 2meV

= 6.63 × 10-34 J.s / 2 × 9.11 × 10-31 × 1.602 × 10-19 × 15 × 103

= 1.0 × 10-11 m = 0.010 nm so proper length in the lab frame.

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