L33_a - HOW DO MAXWELL’S EQUATIONS GIVE RISE TO...

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Unformatted text preview: HOW DO MAXWELL’S EQUATIONS GIVE RISE TO ELECTROMAGNETIC WAVES? (SKETCH ONLY) Faraday’s Law: t d d s d E B 1 loop around Φ − = ⋅ ∫ r r gives 1) (Eq. t B x E ∂ ∂ − = ∂ ∂ Ampere’s Law: t d d s d B E 2 loop around Φ = ⋅ ∫ µ ε r r gives 2) (Eq. t E x B ∂ ∂ − = ∂ ∂ µ ε Take derivatives of Eqs. 1 and 2 with respect to x and t and then combine to get: • Wave equations for E r and B r : 2 2 2 2 t E x E ∂ ∂ = ∂ ∂ µ ε 2 2 2 2 t B x B ∂ ∂ = ∂ ∂ µ ε E B z x y 1 2 SOLUTIONS of 2 2 2 2 t E x E ∂ ∂ = ∂ ∂ µ ε and 2 2 2 2 t B x B ∂ ∂ = ∂ ∂ µ ε • ( ) t x k E E ω − = cos max and ( ) t x k B B ω − = cos max • WAVES with speed m/s 10 99 . 2 1 8 × = = µ ε c • f π ω 2 = determined by source of electromagnetic wave • f c λ = constant for ALL electromagnetic waves • λ π 2 = k From t B x E ∂ ∂ − = ∂ ∂ get max max B E k ω = so that c k E = = ω max max B • so for ALL points on wave, c E = B ENERGY CARRIED BY ELECTROMAGNETIC WAVES (24.5) Power (energy/unit time) carried across a unit area of surface by em wave: • given by Poynting vector S r o magnitude B E S r r r × = 1 µ is power/unit area o direction B E r r × is direction of propagation Average power/unit area is:...
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L33_a - HOW DO MAXWELL’S EQUATIONS GIVE RISE TO...

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