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My question is: in electromagnetic waves, if we consider the electric field as a sine function, the magnetic field will be also a sine function, but I am confused why that is this way.

If I look at Maxwell's equation, the changing magnetic field generates the electric field and the changing electric field generates the magnetic field, so according to my opinion if the accelerating electron generates a sine electric field change, then its magnetic field should be a cosine function because $\frac{d(\sin x)}{dx}=\cos x$.

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    $\begingroup$ "changing magnetic field generates the electric field and the changing electric field generates the magnetic field" - I think this is misleading. Maxwell's equations aren't statements of cause and effect. Although we talk about one field changing inducing another, they happen at the same time. An increasing magnetic field doesn't really cause a curl to exist in the electric field, they are physically the same - an increasing magnetic field cannot exist without the curl in the electric field. $\endgroup$ – andars Feb 18 '19 at 4:46
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    $\begingroup$ It's worth stating clearly that the in-phase nature of the waves is true in the far field (i.e. when the waves are examined much farther from the source than the size of the source), but that this is not the case in the near field (i.e. when you are close to the source). $\endgroup$ – dmckee --- ex-moderator kitten♦ Feb 18 '19 at 5:58
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    $\begingroup$ Note that for standing waves E and B are out of phase. $\endgroup$ – my2cts Feb 18 '19 at 7:28
  • $\begingroup$ @my2cts, i do not believe you meant what you wrote, just think of an open ended (or short circuited) transmission line. $\endgroup$ – hyportnex Feb 18 '19 at 15:18
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E and B are in phase for a running plane wave, but are out of phase for a standing wave. standing waveThis can be easily seen by considering the vector potential, $A(t, x) $. Using $E = \partial_t A$ and $B=\partial_x A$. For $A=sin(\omega t - kx) $ you find that E and B are in phase. For $A=sin(\omega t) sin(kx) $, a standing wave, E and B are out of phase.

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  • $\begingroup$ Thanks. Could you tell me when can standing waves occur ? $\endgroup$ – Bálint Tatai Feb 19 '19 at 14:57
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The Maxwell equations that relate electric and magnetic fields to each other read (in vacuum, in SI units) as \begin{align} \nabla \times \mathbf E & = -\frac{\partial\mathbf B}{\partial t} \\ \nabla \times \mathbf B & = \frac{1}{c^2} \frac{\partial\mathbf E}{\partial t}, \end{align} where the notation $\nabla \times{\cdot}$ is a spatial derivative (the curl). This means that both sides have derivatives, and if you're applying them to a function like $\cos(kx-\omega t)$, then they will both change the cosine into a sine. This is what locks the phase of both waves to equal values.

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  • $\begingroup$ Just to comment with my comment on the questions, this explanation works nicely if spatial arrangement of the fields is uniform enough (as in a plane wave, which is to say in he far-field), but misses important details if the wave has a no-planar structure (as in the near-field). $\endgroup$ – dmckee --- ex-moderator kitten♦ Feb 18 '19 at 6:00
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    $\begingroup$ @dmckee Oh, absolutely. And indeed if you don't have a plane wave (which can mean that you have anything from a near-field to something as sophisticated as the superposition of two plane waves) then the fields are not required to be (strictly) orthogonal to each other or to the propagation direction. Hence the "if..." in this answer. $\endgroup$ – Emilio Pisanty Feb 18 '19 at 7:13
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This is one of those 'why' questions that physics can or cannot answer, depending on what you want from answer to 'why'.

If equations are a satisfactory explanation, then the Maxwells Equation in Emilios answer are a complete answer.

Unfortunately, not far beneath the surface of that answer is 'why do Maxwells Equations' fit reality?' or 'why do fields behave the way they do so that we can derive Maxwells Equations?'. Wigner along with many other physicists was similarly troubled by such questions.

It doesn't get any more intuitive if you go down further to QED to try to explain the classical behaviour.

At the lowest level, the answer is 'that's the way Nature behaves'.

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  • $\begingroup$ Lower level physical theories can describe why, but I don't think any where put forth outside if either theories because at the time of Maxwell the atom hadn't yet been proven or shown necessary. Gravitation and any field theories had this problem. I think in general everyone supposed the mechanisms describing these fields, happen on scales far to small, and for all they knew possibly fundamental to reality. Then quantum came along and were all taught to not try and describe these mechanisms. Simply by the words of Feynman that it is impossible to understand. $\endgroup$ – marshal craft Feb 18 '19 at 7:43
  • $\begingroup$ And here we are today, no further really have we progressed it seems. Further it is maybe just as interesting to ask why nobody considers this? Perhaps for the pure sport of it alone, why nobody has framed wiled hypothesis as to the mechanisms underlying fields? $\endgroup$ – marshal craft Feb 18 '19 at 7:45
  • $\begingroup$ But from an abstract math perspective, chaos, infinitely complicated systems i guess are more interesting. Maybe they all know and don't say, that we need to be able to handle insanely complex, large systems and discover abstract principles which emerge at large scales, to get ahead of the curve. $\endgroup$ – marshal craft Feb 18 '19 at 7:48
  • $\begingroup$ This refusal of even an attempt of a deeper understanding is unsatisfying. A mathematical formula, such as given by Emilio, is nothing but an insight cast into a strict notational form. Only by understanding the formula and the insight it communicates do we obtain a true mental grasp of a matter which allows us to mentally proceed on our own. $\endgroup$ – Peter - Reinstate Monica Feb 18 '19 at 13:45
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The E and H fields in a time-harmonic EM wave are in phase in the time domain when the medium's polarization (electric and magnetic) are in phase with the corresponding fields. You can see that polarization fields inherently act as 'source' terms in Maxwell's equations, and hence, instantaneous polarization implies in-phase relationship. However, whenever there is dissipation (such as existence of conduction current, or out of phase polarization), the E and H fields are no longer in phase. In other words, the one phasor cannot respond instantaneously to the changes of the second one in time. Note that regardless of propagating or standing wave, E and H fields are in phase with each other in the time domain in a lossless medium (for a standing wave, they are 'out of phase' spatially).

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