Notes 7 Spring 2005

Notes 7 Spring 2005 - Notes 7 Spring 2005.doc Conservation...

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Notes 7 Spring 2005.doc Conservation of Power and the Poynting Vector Note the following: () () ( ) A A A A A A A A A A G G G G G G G G G G t t 2 1 t t 2 1 2 1 2 t = = + = Recalling the first two of Maxwell’s equations we can manipulate them as follows: () () () 2 2 1 c 2 1 c c c c c t J t J t J t J t J t J E E E E E E E E D E E D E H E D H ε + = ε + = ε + = + = + = × + = × G G G G G G G G G G G G G G G G G G G G G G G Recalling the vector identity given in Notes 2, i.e., ( ) ( ) ( ) b a a b b a G G G G G G G G G × × × we find that if we let and E G G a H G G b we get () () () () 2 2 1 2 1 t t t t t t H H H H H H H B H E H B E µ = µ = µ = µ = = × = × G G G G G G G G G G G G G G G () ()( H E E H H E G G G G G G G G ) × × = × ; and substituting with the results from above yields () () ( ) ε + µ = × 2 2 1 2 2 1 E E H H E t J t c G G G G G () () ( ) ε + + µ = × 2 2 1 2 2 1 E E H H E t J t c G G G G G () ( µ + ε + = × 2 2 1 2 2 1 H E E H E t J c ) G G G G G 1
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Notes 7 Spring 2005.doc The first term on the right, , has units of c J G G E 3 3 2 m W m VA m A m V = = as must the two derivatives on the right. These are recognized as units of a power density. In particular, is the ohmic loss, c J G G E P (power loss per unit volume), of the electric field and is manifested as heat dissipated to the environment when the medium has a non-zero conductivity (this term is obviously zero for free space). It might be instructive to recall the power relation from
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This note was uploaded on 01/20/2012 for the course EE 4460 taught by Professor Czarnecki during the Fall '10 term at LSU.

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Notes 7 Spring 2005 - Notes 7 Spring 2005.doc Conservation...

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