Maxwell's equations in relativistic physics

[elivil]

Homework Statement

It is often said that Maxwell's equations in differential form hold in special relativity while Maxwell's equations in integral form don't hold. Consider one of equations:
$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$
$$\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {1} {c} \frac {d} {dt} \int \mathbf {B} \cdot \mathrm{d} \mathbf {S}$$

Consider the integral form. It means that if one has an alternating magnetic field, then in any contour around this field circulation of electrical field immediately appears . If one takes a very long tube and at one end of the tube somehow generates an alternating magnetic field then in contour (radius R) around the tube circulation of electric field immediately appears. But in relativistic case it can't appear immediately because it will take time $t=\frac {R} {c}$ for news about alternating magnetic field to come to this contour. So here we have the violation of this equation in relativistic case.
Still this 'paradox' can be solved in terms of classical electrodynamics. But how?

Homework Equations

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$
$$\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac {1} {c} \frac {d} {dt} \int \mathbf {B} \cdot \mathrm{d} \mathbf {S}$$

The Attempt at a Solution

Maybe it takes time for propagation of alternating magnetic field from one end of the tube to the point A where cross-section of the tube by the plane of the contour is. So in fact it takes equal time to propagate inside the tube and outside the tube so it appears as simultaneously arising magnetic field at point A in the tube and electric field in the contour around the tube. But I'm not quite sure.

 

[Chi Meson]

This should be moved to the advanced physics forum. I'd take a stab at it, but my Maxwell is rusty, and I might hurt myself with it.

 

[Moetasim]

Your explanation is on the right track. In special relativity, the concept of simultaneity is relative and depends on the observer's frame of reference. This means that the time it takes for the news of the alternating magnetic field to reach the contour will be different for different observers. However, in classical electrodynamics, the equations are still valid and can be used to predict the behavior of the fields. The apparent violation of the integral form of Maxwell's equations in special relativity can be resolved by considering the time it takes for the information to reach the observer in their specific frame of reference. This concept is known as the "relativistic correction" and is an important aspect of understanding the behavior of electromagnetic fields in special relativity.