FWLec20 - A F Peterson Notes on Electromagnetic Fields...

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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 Fields & Waves Note #20 Energy and the Poynting Vector Objectives: Motivate and develop expressions for the instantaneous and time-average Poynting vectors. Use the Poynting vectors to characterize the power carried by an electromagnetic wave for several examples involving lossless and lossy materials. Just as a static electric or magnetic field acts as a store of potential energy, a time-varying electromagnetic wave carries energy with it. In this note, an expression for the power carried by an electromagnetic wave is derived, and used to investigate several applications. The Poynting Theorem Suppose that the electric field and magnetic field are explicit functions of time, given by e x y z t ( , , , ) and h x y z t ( , , , ), respectively, and that the medium of interest is characterized by permittivity e , permeability m , and conductivity s . To investigate the power carried by an electromagnetic wave, we begin with a mathematical derivation based on several equations. The following relation is a vector identity, meaning that it is satisfied by any vector functions e and h : ¥ = — ¥ - — ¥ ( ) e h h e e h (20.1) However, the fields satisfy Faraday’s Law — ¥ = - e h t m (20.2) and Ampere’s Law — ¥ = + h e t e e s (20.3) which can be substituted into (20.1) to obtain ¥ = - Ê Ë Á ˆ ¯ ˜ - + Ê Ë Á ˆ ¯ ˜ = - - - ( ) e h h h t e e t e h h t e e t e e m e s m e s (20.4) The expression in (20.4) can be rewritten after recognizing that
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A. F. Peterson: Notes on Electromagnetic Fields & Waves 11/04 1 2 1 2 1 2 d dt h h h dh dt dh dt h h dh dt ( ) = + = (20.5) to produce ¥ = - ( ) - ( ) - = - - - ( ) e h t h h t e e e e t h t e e m e s m e s 2 2 2 2 2 2 2 (20.6) We are concerned with the power stored by the field in some region of space and the power flowing out of that region. Therefore, we integrate both sides of (20.6) over a volume V : ¥ = - - - Ï Ì Ó ¸ ˝ ˛ ÚÚÚ ÚÚÚ ( ) e h dv t h t e e dv V V m e s 2 2 2 2 2 (20.7) The left-hand side of this expression can be simplified using another vector identity known as the divergence theorem : = ÚÚÚ ÚÚ A dv A dS V S (20.8) where S is the surface enclosing the volume V . Equation (20.8) holds for any vector A , and is essentially a generalization of the integration-by-parts formula to the vector situation. Therefore, the left-hand side of (20.7) can be rewritten as ¥ = ¥ ÚÚÚ ÚÚ ( ) e h dv e h dS V S (20.9) By combining this with the right-hand side of (20.7), we obtain e h dS t h t e e dv S V ¥ = - - - Ï Ì Ó ¸ ˝ ˛ ÚÚ ÚÚÚ m e s 2 2 2 2 2 (20.10) This result is known as the Poynting Theorem . It was developed in 1883 by J. H.
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