P214soln7 - Problem set 7 solutions by Ali Vanderveld (1)...

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Unformatted text preview: Problem set 7 solutions by Ali Vanderveld (1) The electric field is ~ E ( ~ r,t ) = < "- d ~ A dt # = < h ziAe i ( ~ k ~ r- t ) i . (1) Since the electric field is pointing solely in the z direction, Gausss law will be satisfied if E z z = 0 , (2) which means that E z must not be a function of z ; this will happen if ~ k does not have a z component, i.e. ~ k = ( k 1 ,k 2 , 0). In order to find the dispersion relation, plug ~ E into the wave equation 2 ~ E = 1 c 2 2 ~ E t 2 (3) to see that q k 2 1 + k 2 2 = ~ k = c . (4) Also, this wave is propagating in the ~ k direction, and it is polarized in the z direction. (2) We are given the vector potential, which has y and z components and thus is only a function of x position ~ A ( x,t ) = ( y + e i z ) Ae i ( kx- t ) , (5) where the physical electric and magnetic fields are ~ E ( x,t ) = < "- d ~ A dt # (6) and ~ B ( x,t ) = < h ~ ~ A i , (7) respectively. To this end, we can calculate d ~ A dt =- i ~ A =- ( y + e i z ) iAe i ( kx- t ) (8) 1 and ~ ~ A = A y x z- A z x y = ( z- e i y ) ikAe i ( kx- t ) (9) such that ~ E ( x,t ) = < ( y + e i z ) iAe i ( kx- t ) =- A { y sin( kx- t ) + z [sin cos( kx- t ) + cos sin( kx- t )] } =- A [ y sin( kx- t ) + z sin( kx- t + )] (10) and ~ B ( x,t ) = < ( z- e i y ) ikAe i ( kx- t ) = kA { y [sin cos( kx- t ) + cos sin( kx- t )]- z sin( kx- t ) } = kA [ y sin( kx- t + )- z sin( kx- t )] . (11) Notice that B y =- k E z (12) and B z = k E y ; (13) using this information, we can then calculate the dot product ~ E ~ B = E y B y + E z B z = E y- k E z + E z k E y = 0...
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P214soln7 - Problem set 7 solutions by Ali Vanderveld (1)...

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