Maxwell's equations in relativistic case

[elivil]

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 the 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?

 

[Vailanor]

The solution to this 'paradox' can be found in classical electrodynamics. According to the Faraday's law of induction, the electric field induced within the contour is proportional to the rate of change of the magnetic flux through the contour. So, in the case of an alternating magnetic field, the electric field will be induced with a delay of t=\frac {R} {c}. This means that the integral form of Maxwell's equations holds in special relativity, since the electric field will be induced with a delay of t=\frac {R} {c}, in accordance with the speed of light.

 

[astrokreng]

I would respond by saying that the concept of time and simultaneity in special relativity is different from classical mechanics. In special relativity, the speed of light is constant and the laws of physics are the same in all inertial frames of reference. This means that the concept of "immediately" is not applicable in the same way as it is in classical mechanics.

In the example given, the circulation of electric field in the contour around the tube may not appear immediately, but this does not violate Maxwell's equations. The integral form of Maxwell's equations takes into account the time it takes for information to travel from one point to another.

Additionally, the concept of causality in special relativity also plays a role. In the example, the alternating magnetic field at one end of the tube is the cause for the circulation of electric field in the contour. However, in special relativity, causality is preserved, meaning that the cause must always precede the effect. This means that the circulation of electric field may not appear immediately, but it will still be caused by the alternating magnetic field.

In conclusion, the apparent violation of Maxwell's equations in the relativistic case can be explained by understanding the differences in the concept of time and causality in special relativity. Furthermore, the equations still hold true, but they must be interpreted differently in the relativistic context.