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Maxwell's curl equation for electric field says that time varying magnetic field produces electric field instantaneously but nothing can happen instantaneously according to special relativity. What is the mistake in my understanding since Maxwell's equations are accepted?

Since nothing can happen instantaneously so how is changing magnetic field producing the electric field instantaneously in EM wave according to $$E(x,t) = E_\text{max} \cos(kx - \omega t+\phi)$$
$$B(x,t) = B_\text{max}cos(kx - ωt + \phi)$$ which shows that they are in phase(means no delay)? I feel there should be some delay in wave of electric field as nothing is instantaneous( vice versa if changing electric field).

Please give an intuitive understanding. Thanks in advance.

Edit: the argument I am giving here is different from the question mentioned in possible duplicate. I am asking about instantaneous production of field and not about how they have same expressions.

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marked as duplicate by garyp, John Rennie electromagnetism Jul 9 at 11:52

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ It might be helpful to think carefully about why you think they should be out of phase. $\endgroup$ – garyp Jul 9 at 10:49
  • $\begingroup$ @garyp my argument is different from the mentioned question I would better change the question although they seem similar $\endgroup$ – Trilok Girish Kamagond Jul 9 at 11:03
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    $\begingroup$ "Maxwell's curl equation for electric field says that time varying magnetic field produces electric field instantaneously" This is the mistake. There is no production per se: electric and magnetic fields are two manifestations of the electromagnetic tensor. What the equations say is more that a variation of one field is $accompanied$ by (a curl) of the other field; but this only describes the relation between the different element of a unique entity, the electromagnetic tensor, without a cause and effect relation. See en.wikipedia.org/wiki/Jefimenko%27s_equations#Discussion $\endgroup$ – David Jul 9 at 11:15
  • $\begingroup$ @John Rennie I feel the argument in that question and this question are different. Although both of question asks about phase. $\endgroup$ – Trilok Girish Kamagond Jul 9 at 12:05
  • $\begingroup$ @David thank you so much $\endgroup$ – Trilok Girish Kamagond Jul 9 at 12:07
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The simplest way to show the problem in the argument is by looking carefully at Maxwell's equations. Namely they are "laws" that apply at a given point in space and time (or space-time) so there is no violation of relativity because the "information" has to travel zero distance.

Updated Theory Viewpoint:
Moreover the idea that magnetic and electric fields are different fields is historically correct but we understand nowadays that they are both manifestations of a single spin 1 gauge field (the photon field) in particular reference frames, so one could say the behaviour we observe (induction in this case) is already encoded in the laws governing such field, that is: its spin 1 structure (and massless-ness) immediately lead to Maxwell's equations.

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