Phys4D - Electromagnetic waves Maxwell equations: B j E t...

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Unformatted text preview: Electromagnetic waves Maxwell equations: B j E t (1) E B t (2) E (3) B (4) By applying derivative t to Eq. (1) and to Eq. (2) we obtain: E j t 2 E t 2 (5) Vector algebra: a ( b c ) b ( a c ) c ( a b ) Similarly ( E ) ( E ) 2 E (6) By using (6) and (3) Eq. (5) can be written in the form: 2 E 2 E t 2 j t 1 (7) The first term in l.-h. side of (7) contains the vector Laplace operator. In rectangular (cortesian) coordinates it is given by 2 E 2 E x x 2 2 E x y 2 2 E x z 2 i 2 E y x 2 2 E y y 2 2 E y z 2 j 2 E z x 2 2 E z y 2 2 E z z 2 k i 2 E x j 2 E y k 2 E z Outside of the region with sources Eq. (7) is reduced to the wave equation 2 E 2 E t 2 (8) For 1-d case E 0, E y ( x ),0 in vacuum (8) is reduced to 2 E y x 2 2 E y t 2 (9) Let us assume that the solution of Eq. (9) has the form E y ( x , t ) f ( x Ut ) (10) The factor v is a constant. The function f can be any function of a single variable. The purpose of writing E y ( x , t ) as we have in (10) is to make the waveform move as a unit in the positive- x direction as time passes. We know that if f ( x ) is any function of x , then f ( x x ) is the same function, shifted to the right a distance x along the x axis. If instead of f ( x x ) we write f ( x Ut ) , then the function is shifted to the right a distance Ut . This distance increases as time increases, so the function is displaced steadily further out the...
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Phys4D - Electromagnetic waves Maxwell equations: B j E t...

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