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Physics 106b – Problem Set 11 – Due Mar 2, 2005Version 1February 6, 2005These problems cover the material on special relativity in Hand and Finch Chapter 12 and Section6.1 of the lecture notes.This problem set is due on Wed Mar 2 at 5 pm (1 week before the last day of classes) at 311 Downs.It will be worth 50% of a normal problem set. Late problem sets may be turned in up to 1 weeklate for 50% credit as usual. You have plenty of lead time on this problem set, the fact that it isdue one day after the Ph125 set will not be accepted as a mitigating factor.The original intention was to include these on the first E & M problem set, but that did not happen.You no doubt will be unhappy about this extra work, but it would be negligent to have assignedno problems on special relativity – as we always say, the only way to understand the material is todo problems. You’ll be thankful if you take the physics GRE1or any kind of serious astrophysicsor particle/nuclear physics classes (and certainly if you take relativistic QM).I will not be holding regular offic e hours, but feel free to arrange a special app ointment with me ifyou need help with the material.1. Hand and Finch 12.6 (relativistic acceleration). This problem is very useful for calculat-ing the motion of a charged particle in various accelerators (cyclotron, synchrotron, linearaccelerator). In addition to what is asked, do the following:(a) Show that|aµ|2=γ6γ2⊥d~βdt2whereγ2⊥=1 − β2⊥−1=1 −β2−~β ·d~βdtd~βdt2−1is the γ factor due to the lab-frame velocity perpendicular to the lab-frame acceleration.(It’s less complicated than it looks.)(b) Show that the above formula reduces to the two that you found for circular m otion andparallel acceleration.1see, for example, http://128.148.60.98/physics/userpages/students/Artur Adib/gre.html or justgo ogle “physics GRE relativity”)1Notes:• As in the definition of four-velocity, you will find it necessary to convertddτtoddttoobtain the desired explicit form. In our demonstration that uµ= γ1,~β, we use dτ =p|xµ|2and took the derivative with respect to t. We implicitly assumed there that~β was constant. Since we are no longer making that as sumption, the original relationfor τ (t) may no longer hold. What we can be sure of, though, is that the infinitesimalversion of the relation holds:(dτ)2= (dt)2− (d~x · d~x)2dτ = dtq1 − β2=dtγwhere~β may now be a function of time. You will want to use the differential relation.2• There is a minor error in the problem: it should ask you to demonstrate that “therelation betwee n the laboratory acceleration of a particle undergoing circular motion atconstant speed and the acceleration in the instantaneous rest frame of the particle isarest= γ2alab.” That this is a typo is confirmed by Goldstein derivation 7.7 and thegeneric formula for |aµ|2given above.2. Hand and Finch 12.17 (Compton scattering). Notes:(a) In class we have not used the notation kµkµor pµpµ. In general, for any four-vector,the expression aµaµis the invariant norm |aµ|2= (a0)2− (a1)2− (a2)2− (a3)2. Theformal meaning of lowered indices is discuss ed in the lecture notes, but you don’t nee dto know that for this problem.(b) T he kµdefined in this problem is different from the kµdefined in class by a factor ¯h. Youcan basically forget about the kµdefined in class when doing this problem – just take itas given that kµis the four-momentum of the photon, with the time component beingthe energy and the space components being the spatial momentum, and that |kµ|2= 0because the photon res t mas s vanishes.(c) A “backscattered” photon is one with outgoing angle θ0= π.3. You should look at and understand conceptually how to do Hand and Finch 12.4 (the pole-in-barn problem), but it is not to be turned in. This is a classic “relativity of simultaneity”problem that everyone ought to know how to do.2Two finer points: 1) with the differential relation, we could of course have derived the four-velocity ex press ionwithout the assumption of constant velocity; hence, it holds even when the velocity is not constant. 2) If we integratethe differential relation, we findτ (t) =Zt0d˜tq1 −β(˜t) 2which may in general be different fromp|xµ|2=√t2− ~x · ~x, hence the distinction between using the differential andintegral


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CALTECH PH 106B - Problem Set 11

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