Saturday, 27 January 2018

electromagnetic radiation - How does energy transfer between B and E in an EM standing wave?


I'm trying to understand how an electric field induces a magnetic field and vice versa, its associated energy, as well as relating it to my understanding of waves on a string.


Using a standing wave as an example, I came up with the equations


E=Asin(ωt)sin(kx)ˆy


B=Akwcos(ωt)cos(kx)ˆz


I checked them against Maxwell's equations, and they're self-consistent. At time 0, this reduces to:


B=Akwcos(kx)ˆz


Since the electric field is 0, based on the Poynting vector, there's no energy transfer at this time. At this time, at a node where B=0, there's neither electric field nor magnetic field. If there's no energy transfer, and no energy stored in either field, then how can an electric field exist at this point at some time later? How is the energy stored, or transferred from elsewhere?



Answer



The energy conservation is written ut+divS=0, where u is the energy density E2+B2, and S is the Poynting vector EB (skipping irrelevant constant factors).



If you choose x,t such as sinωt=0 and coskx=0, both E, B, the energy density u and the Poynting vector S will be zero.


We have Ssin2kxsin2ωt ˆx. The divergence of the Poynting vector will be divScos2kxsin2ωt, so it it zero too (because of the t dependence ), and so ut=0. However, the first time derivative of div(S)is not zero : (divS)tcos2kxcos2ωt, so, from the energy conservation equation, the second derivative of the density energy 2ut2=(divS)t is not zero.


So, you may write :


u(x,t+ϵ)=ϵ222ut2+o(ϵ2)


So, at infinitesimal times after t, the energy density is not zero.


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