As I have read that "time varying electric field is source of changing magnetic field and time varying magnetic field is source of changing electric field "
Hence, I have the following doubt : If time varying nature of one field originates other changing field. Then, why are they in phase?
One is changing and producing the second thing hence second must depend upon first's derivative, I. e. If one is sine then second will be cosine Hence a phase difference of pi/2 and not in phase. Where am I wrong?
Answer
It is not correct to say that a changing E-field "generates" a changing B-field, and vice-versa.
Maxwell's equations imply that they co-exist. The presence of a changing E-field means that there must be an accompanying B-field.
How and why they are in phase can be seen by solving Maxwell's equations in vacuum. The solutions have the form $$\vec{E} = \vec{E_0} f(\vec{k}\cdot \vec{r} - \omega t),$$ where $\omega/k=c$ and $\vec{E_0}\cdot \vec{k}=0$.
If you take the curl of this field (the right hand side of Faraday's law), you take spatial first derivatives. The corresponding B-field is then found by integrating this with respect to time. $$\vec{B}= -\int \nabla \times \vec{E}\ dt$$
The process of differentiating with respect to spatial coordinates and then integrating with respect to time, ensures that the function $f$ and it's argument remain unchanged and therefore the E- and B-field must be in phase.
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