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Phys4D

# Phys4D - Electromagnetic waves Maxwell equations B j E(1 B...

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Electromagnetic waves Maxwell equations:   B j  E t (1)   E   B t (2)   E (3)  B 0 (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

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Outside of the region with sources Eq. (7) is reduced to the wave equation 2 E  2 E t 2 0 (8) For 1-d case E 0, E y ( x ),0 in vacuum (8) is reduced to 2 E y x 2 0 0 2 E y t 2 0 (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 0 ) is the same function, shifted to the right a distance x 0 along the x axis. If instead of f ( x x 0 ) 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 x axis. The displacement is by a distance Ut , which means that the velocity of motion is U . It is easy to see that the entire
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Phys4D - Electromagnetic waves Maxwell equations B j E(1 B...

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