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Find the momentum and the velocity acquired by an electrically charged particle which has travelled through a potential difference varphi =varphi_(1) - varphi_(2). Take the initial velocity of the particle to be zero. Do the calculation both for the nonrelativistic and the relativistic cases. |
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Answer» Solution :Consider the PROBLEM using the approximation of Newto nian mechanics. Put `K_(0) = 0, varphi_(1) -varphi_(2)= varphi, A = K- K_(0) = q(varphi_(1) -varphi_(2))` `qvarphi= (1)/(2)mu^(2) , U= SQRT(2qvarphi//m),p= mu= sqrt(2mqvarphi)` (b) In the relativistic case `K= qvarphi= sqrt(p^(2)c^(2)-E^(2))-E_(0)` we obtain `p= (1)/(c) sqrt(qvarphi(2E_(0)+qvarphi)), u= (m uc^(2))/(mc^(2) ) = (pc^(2))/(E)= (csqrt(qvarphi(2E_(0)+qvarphi))/(E_(0)+qvarphi))` In the Newtonian approximation `E_(0) gtgtqvarphi`and the formulas assume the form `p~~ (1)/(c)sqrt(2E_(0)qvarphi)= sqrt(2m_(0)qvarphi), u= (csqrt(2E_(0)qvarphi)/(E_(0)))= sqrt((2qvarphi2)/(m_(0)))` We have of COURSE obtained the same expressions as in (a), for in that case `m= m_(0)` |
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