# Maxwell's equations in relativistic physics

• elivil
In summary, the conversation discusses the differences between Maxwell's equations in differential and integral form, and how the integral form appears to be violated in the relativistic case due to the time it takes for information about an alternating magnetic field to propagate. The solution to this 'paradox' is proposed to be that the propagation of the magnetic field takes equal time inside and outside of the tube, resulting in the simultaneous appearance of the magnetic field at a point in the tube and the electric field in a contour around the tube. However, further discussion on this topic is suggested to be moved to the advanced physics forum.
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.

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.

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.

## 1. What are Maxwell's equations in relativistic physics?

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields in the presence of charges and currents. In relativistic physics, these equations are modified to account for the effects of Einstein's theory of special relativity.

## 2. How do Maxwell's equations change in relativistic physics?

In relativistic physics, the equations are modified by introducing the concept of a four-dimensional spacetime, where time and space are combined. This results in the addition of a new term, called the displacement current, in one of the equations. Additionally, the equations are written in a covariant form to account for the effects of special relativity.

## 3. What is the significance of Maxwell's equations in relativistic physics?

Maxwell's equations play a crucial role in understanding the behavior of electromagnetic fields in the presence of relativistic effects. They have been extensively tested and are considered to be one of the most successful theories in physics, as they accurately describe a wide range of phenomena.

## 4. How do Maxwell's equations relate to other theories in physics?

Maxwell's equations are a cornerstone of classical electromagnetism and have been instrumental in the development of other theories such as quantum mechanics and general relativity. They also provide a link between the fields of electromagnetism and optics, as they explain the behavior of light as an electromagnetic wave.

## 5. What are some practical applications of Maxwell's equations in relativistic physics?

Maxwell's equations have a wide range of practical applications, including the design of electronic devices, such as radios and televisions, the understanding of electromagnetic radiation and its effects on biological systems, and the development of technologies such as MRI machines and particle accelerators. They have also been used in the development of the GPS system and other technologies that rely on electromagnetic waves.

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