how can 2 EM waves (travelling in opposite directions) null each other at a point in space but continue to propagate beyond the point in space where they interact to null each other?
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1$\begingroup$ Related: physics.stackexchange.com/q/246808/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Apr 2, 2016 at 19:54
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1$\begingroup$ @Qmechanic essentialy a duplicate, isn't it? $\endgroup$– AccidentalFourierTransformCommented Apr 2, 2016 at 20:16
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$\begingroup$ @AccidentalFourierTransform It's 100% not a duplicate, it's as different as can be. A string has position and velocity as independent parameters because Newton is second order. Whereas Maxwell is first order so knowing the complete values of the electromagnetic field at one snapshot in time tells you what the time derivatives are. You aren't free to independently specify the time rate of change. Despite all the induction blather about time derivatives of fields causing values of the fields, it actually is that the values of electromagnetic fields determine the time derivatives. $\endgroup$– TimaeusCommented Apr 3, 2016 at 2:13
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$\begingroup$ @Timaeus I see your point, but I'd say it's 50% a duplicate because it is the same as the other post if we change $y(x,t)\leftrightarrow \boldsymbol E(x,t)$, $\dot y(x,t)\leftrightarrow \boldsymbol B(x,t)$. Or put it another way: Maxwell's equations can always be written as $(\partial_t^2-\nabla^2)A_\mu=0$ for $A_\mu$ the vector potential, so we do have a second order wave equation. EDIT: now I realise this is precisely what your answer below is saying, so we agree there is a resemblance to the other post, right? $\endgroup$– AccidentalFourierTransformCommented Apr 3, 2016 at 9:49
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$\begingroup$ @AccidentalFourierTransform If you want to answer it with the linked question you could make it be about all the A and then if A is momentarily zero everywhere you could ask why it isn't gone. But that's just the case when the magnetic field is momentarily gone and the "kinetic" term for A just means there is a nonzero electric field at the moment the magnetic field is gone. The real answer is that two waves can't cancel electrically and magnetically and have opposite momentum. So the situation described doesn't happen. $\endgroup$– TimaeusCommented Apr 3, 2016 at 18:18
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You describe an impossible scenario.
If two electromagnetic waves travel in opposite directions and their electric fields point in opposite directions, then their magnetic fields point in the same direction.
If two electromagnetic waves travel in opposite directions and their magnetic fields point in opposite directions, then their electric fields point in the same direction.
More generically, when the electric and magnetic fields both pick up a minus sign, the Poynting vector remains the same. So the momentum density can't point in opposite directions when the electric and magnetic parts of the electromagnetic field are both opposite.
If you want to get super technical: a zero field can satisfy all these things. But then there are no waves and no energy and no dynamics.
