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 )