Chapter.1page27

Chapter.1page27 - T / 2 T / 2 1 ) and w ( r ) = 1 ( 2 ) 2 e...

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In the cw mode the average power delivered to the currents by the fields is w ( r ) = 1 T T / 2 T / 2 E ( r , t ) J ( r , t ) dt = 1 ( 2 ) 2 ∫∫ −∞ e ( r , ) j ( r , ′′ ) 1 T T / 2 T / 2 exp [− i ( + ′′ ) t ] dtd d ′′ Let = ( + ′′ ) = 1 ( 2 ) 2 ∫∫ −∞ e ( r , ) j ( r , ) 1 T 1 i [ exp (− i T 2 )− exp ( i T 2 )] d d = 1 ( 2 ) 2 ∫∫ −∞ e ( r , ) j ( r , ) sin [ T / 2 ] T / 2 d d , 112 In the cw mode the frequency dependent fields and sources should be sharply peaked around the selected frequency, say 0 . The time average will be taken over a time, T = 2 / 0 , which is short compared to 2 / Δ where Δ is the frequency spread of the fields and sources, = 0 ± Δ In the integral over the value of will be restricted to be near 0 by the field e ( r , ) . In that case j ( r , ) will also be peaked around = 0 ± Δ and will be restricted to be less than Δ . It follows that T Δ T = Δ 2 / 0 1 (so that sin
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Unformatted text preview: T / 2 T / 2 1 ) and w ( r ) = 1 ( 2 ) 2 e ( r , ) j r , d d = 1 ( 2 ) 2 e ( r , ) d j r , d The w ( r ) is real and can be rewritten, w ( r ) = 1 ( 2 ) 2 [ e ( r , ) + e ( r , ) ] d [ j r , + j r , ] d = 1 ( 2 ) 2 2Re e ( r , ) d 2Re j r , d = 1 ( 2 ) 2 2Re [ e ( r , ) d j r , d ] w ( r ) = 1 ( 2 ) 2 e ( r , ) d j ( r , ) d + c . c . 113...
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This note was uploaded on 12/19/2010 for the course PHYS 423 taught by Professor G. during the Spring '10 term at Missouri S&T.

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