329lect20 - 20 Poynting theorem and monochromatic waves •...

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Unformatted text preview: 20 Poynting theorem and monochromatic waves • The magnitude of Poynting vector S = E × H represents the amount of power transported — often called energy flux — by electromagnetic fields E and H over a unit area transverse to the E × H direction. This interpretation of the Poynting vector is obtained from a conservation law extracted from Maxwell’s equations (see margin) as follows: ∇ · D = ρ ∇ · B = 0 ∇ × E =- ∂ B ∂t ∇ × H = J + ∂ D ∂t . 1. Dot multiply Faraday’s law by H , dot multiply Ampere’s law by E , ( ∇ × E =- ∂ B ∂t ) · H ( ∇ × H = J + ∂ D ∂t ) · E and take their difference: H · ∇ × E- E · ∇ × H =- ∂ D ∂t · E- ∂ B ∂t · H- J · E . ∇ · ( E × H )- ∂ ∂t ( 1 2 E · E + 1 2 μ H · H ) 2. After re-arrangements shown above, the result can be written as 1 ∂ ∂t ( 1 2 E · E + 1 2 μ H · H ) + ∇ · ( E × H ) + J · E = 0 . • Poynting theorem derived above is a conservation law just like the continuity equation ∂ρ ∂t + ∇ · J = 0 : Poynting theorem – The first term on the left, ∂ ∂t ( 1 2 E · E + 1 2 μ H · H ) , is time rate of change of total electric and magnetic energy den- sity....
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329lect20 - 20 Poynting theorem and monochromatic waves •...

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