How to derive the energy of a electromagnetic plane wave

Saturday, March 10, 2018 at 04:51

Energy of a E-M plane wave on a detector:
ε=∫Pdtε = ∫Pdtε=∫Pdt
∵For a plane wave, S is constant over the area, so: 
P=SA=(EH)A=(E(ϵcE))A=(ϵcE²)A∴ε=∫Pdt=∫(ϵcE²)A⋅dtP =SA=(EH)A=(E(ϵcE))A=(ϵcE²)A \\
∴ε = ∫Pdt = ∫(ϵcE²)A·dtP=SA=(EH)A=(E(ϵcE))A=(ϵcE²)A∴ε=∫Pdt=∫(ϵcE²)A⋅dt
 ∵These are constants 
∴ε=∫(ϵcE²)A⋅dt=(ϵE²)∫A⋅c⋅dt∴ε = ∫(ϵcE²)A·dt = (ϵE²) ∫A·c·dt ∴ε=∫(ϵcE²)A⋅dt=(ϵE²)∫A⋅c⋅dt
∵·c·dt is the length and A is the area 
∴∫A·c·dt gives a volume 
∴ε=ϵE²⋅V∴ε = ϵE²·V∴ε=ϵE²⋅V