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Unformatted text preview: Modern Physics Solutions Solution 1 To find the fraction of particles which reach the detector, we have to find the time that it takes for particles to reach the detector. Let be the number of the generated particles and N be the number of particles which reach the detector. The relation between them is given by: N = t N N exp . ( 1 ) Eq.1 is the formula which is used to find the number of particles after time t when their life time is . Therefore we have to find t and . The generated particles have a life time 100 = ns, in their rest frame. This means that if we travel with the particle the particle will decal after 10ns. But this is not the time which is measured in the laboratory. Since the rest frames of the particles move with respect to the lab-frame. According to the special relativity there is dilation of time between the time measured in the lab-frame and the one in the rest frame of the particle. The measured time is related to by: 2 2 1 c v = = , ( 2 ) where v is the velocity of the particles. Here the velocity of the particles is unknown. To find the velocity we use the information given in the problem. We know the total energy of the particle. The relation between the total energy and the velocity is given by: 4 2 2 2 2 c m c p E + = ( 3 ) but, v m p = 4 2 2 2 2 2 c m c m E + = [ ] 2 2 2 2 2 c m c v + = using the definition of 4 2 2 2 1 1 c m c v = 4 2 2 c m = or ( 5 ) 2 c m E = The total energy is: 100 100 2 = = c m E (6) Substitute Eq.6 in Eq.2 we obtain: s 8 10 = . ( 7 ) From Eq.6 and the definition of we get: c v ( 8 ) The time for particles to reach the detector is given by:...
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