HW08'sol - 1 C\User\Teaching\505-506\S11\HW08'sol PHY 506...

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Unformatted text preview: 1 C:\User\Teaching\505-506\S11\HW08'sol.doc PHY 506 Classical Electrodynamics II Spring 2011 Homework 8 with solutions Problem 8.1 (10 points). A charged, relativistic particle has been injected into a uniform electric field which oscillates in time with frequency . Calculate the time dependence of the particle’s velocity (as observed from the lab frame). Solution : Directing axis x along the electric field, and axis y perpendicular to it, but within the plane of the initial velocity of the particle (so that z 0), we can write the relativistic equation (9.144) of its motion, in the lab frame, as , ) ( ) ( , cos ) ( ) ( t d d t t d d y x where mc qE , E being the electric field amplitude. These equations may be readily integrated: . , sin y y x x t (*) Since , 1 1 1 1 2 2 2 2 y x i.e. , 1 1 2 2 2 y x i.e. 2 2 2 ) ( ) ( 1 y x , we can now calculate the Lorentz parameter as a function of time: 2 / 1 2 2 2 / 1 2 2 sin 1 ) ( ) ( 1 y x y x t . (**) Now we can combine Eqs. (*) and (**) to find both reduced velocities as explicit functions of time: ,...
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HW08'sol - 1 C\User\Teaching\505-506\S11\HW08'sol PHY 506...

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