Special relativity - angles between fields

In summary, the value of theta in an electromagnetic field is invariant for all inertial observers and can only be either pi/2 or 3pi/2 when the electric and magnetic fields are perpendicular. Although a rigorous proof is required, a possible approach is to use the invariance of the dot product of the electric and magnetic fields and align the x-axis along the electric field and the y-axis in the direction of the magnetic field. However, a conclusive proof has not yet been found.
  • #1
Aleolomorfo
73
4

Homework Statement


In an electromagnetic filed, the elctric field ##\vec{E}## forms an angle ##\theta## with the magnetic field ##\vec{B}##, and ##\theta## is invariant for all inertial observers. Finding the value of ##\theta##.

Homework Equations


Tranformations of fields perpendicular to the boost:
$$\vec{E'}=\gamma(\vec{E}+(\vec{v}\times\vec{B})_\perp)$$
$$\vec{B'}=\gamma(\vec{B}-(\vec{v}\times\vec{E})_\perp)$$

The Attempt at a Solution


I know that the only ##\theta## that is invariant in ##\frac{\pi}{2}## or ##\frac{3\pi}{2}##, so when the fields are perpendicular. I have always showed it in this way: I use the invariance of ##\vec{E}\cdot\vec{B}=\vec{E'}\cdot\vec{B'}##, so ##EB\cos{\theta}=E'B'\cos{\theta'}##. If ##\theta'=0##, ##\theta## must also be ##0## to hold the equality. Consequently if they are perpendicular in one frame, they are perpendicular in all the other, otherwise the invariance of ##\vec{E}\cdot\vec{B}## does not hold.
I know this is not the best proof, but I think it is in some way logically. However, the exercise wants a rigorous proof.
I have tried using ##\vec{E}\cdot\vec{B}## better but it is a vicious circle. I have aligned the x-axis along ##\vec{E}## and y-axis in order to have the two vectors in the ##xy## plane.
$$\vec{E}=E\vec{x}$$
$$\vec{B}=B(\cos{\theta}\vec{x}+\sin{\theta}\vec{y})$$
I have written ##E'## and ##B'## using the fields-transfromations for a boost along z:
$$E'_x=\gamma(E-vB\sin{\theta})$$
$$E'_y=\gamma v B\cos{\theta}$$
$$B'_x=\gamma B\cos{\theta}$$
$$B'_y=\gamma(B\sin{\theta}-vE)$$
But then everything I have done is inconclusive. I do not know how to conclude the demonstration.
 
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  • #2
Aleolomorfo said:
I know this is not the best proof
Why? It uses invariants. As physicists we love invariants.
 

Related to Special relativity - angles between fields

1. What is special relativity?

Special relativity is a theory in physics that describes the relationship between space and time. It explains how the laws of physics are the same for all observers in uniform motion, and how the speed of light in a vacuum is constant for all observers.

2. What are the angles between fields in special relativity?

In special relativity, the angles between fields refer to the angles between the direction of motion and the direction of an electromagnetic field, such as light. These angles can change depending on the relative velocity between the observer and the source of the field.

3. How does special relativity affect the perception of time?

Special relativity states that time is relative and can appear to pass differently for observers in different reference frames. This is known as time dilation and is caused by differences in speed and gravity.

4. Can special relativity explain the bending of light around massive objects?

Yes, special relativity can explain the bending of light around massive objects, such as stars. This effect, known as gravitational lensing, is caused by the curvature of space-time around massive objects, as predicted by Einstein's theory of general relativity.

5. Is special relativity still considered a valid theory today?

Yes, special relativity is still considered a valid theory today and has been extensively tested and confirmed through experiments and observations. It is an essential part of modern physics and is used in many practical applications, such as GPS systems and particle accelerators.

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