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# Poynting Vector

Module by: Paul Padley. E-mail the author

Summary: The Poynting vector is defined

## Poynting Vector

Now we want to calculate the power crossing a given area A A . During a time Δ t Δ t an EM wave will pass an amount of energy through A A of u c Δ t A u c Δ t A where u u is the energy density of the wave. If we want the power/ m 2 m 2 then we must divide by Δ t A Δ t A . Thus we get S = u c Δ t A Δ t A = u c = 1 μ 0 B 2 c = 1 μ 0 B E c c = 1 μ 0 B E S = u c Δ t A Δ t A = u c = 1 μ 0 B 2 c = 1 μ 0 B E c c = 1 μ 0 B E Now we make the reasonable assumption that the energy flows in the direction of the wave, ie. perpendicular to E E and B B so we can define a vector that has the power per unit area: S = 1 μ 0 E × B S = 1 μ 0 E × B or S = c 2 ε 0 E × B S = c 2 ε 0 E × B This is the Poynting vector. Thus for a plane EM wave we have three useful things E ( r , t ) = E 0 e i ( k r ω t ) E ( r , t ) = E 0 e i ( k r ω t ) B ( r , t ) = B 0 e i ( k r ω t ) B ( r , t ) = B 0 e i ( k r ω t ) and S = c 2 ε 0 E 0 × B 0 e i 2 ( k r ω t ) S = c 2 ε 0 E 0 × B 0 e i 2 ( k r ω t )

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