# Lorentz transformation where electric field vanishes

• castlemaster
In summary, the problem involves finding the velocity of a reference frame in which only E exists in a homogeneous electromagnetic field with E · B = 0 and E ≠ cB. Using the transformations for E and B, and setting B' = 0, we can determine that the velocity must be perpendicular to both E and B. By setting v = s E x B and solving for s, we can find the minimum velocity needed to make B' vanish.
castlemaster

## Homework Statement

We have an homogeneus electromagnetic field with $$E \bullet B=0$$ and $$E \neq cB$$
Find the velocity of the reference frames in which ony E exists.

## Homework Equations

$$\mathbf{E}' = \gamma \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right ) - \left (\frac{\gamma-1}{v^2} \right ) ( \mathbf{E} \cdot \mathbf{v} ) \mathbf{v}$$

$$\mathbf{B}' = \gamma \left( \mathbf{B} - \frac {\mathbf{v} \times \mathbf{E}}{c^2} \right ) - \left (\frac{\gamma-1}{v^2} \right ) ( \mathbf{B} \cdot \mathbf{v} ) \mathbf{v}$$

## The Attempt at a Solution

I guess I can't use the transformations for a boost in the x direction, so I guess I have to use the fact that

$$E \bullet B$$
$$E^2-B^2$$

are invariants under lorentz transformations.
But I don't know how to start. Do I need the EM field tensor for something?

Thanks

First, if you don't set c = 1, your second invariant has a factor of c² missing (I let you find out where). But you won't need them for the solution to your problem anyway.

For this, you have already your transformation formulae for E and B, so just set B' = 0 ...

Hi,

Should I use the fact that E.B=E'.B'=0 and merge the equations from E' and B'?
Cause I don't see a simple way of taking the velocity out of there.

Thanks

Since you know that E.B = 0, surely, as an ansatz, v.B = 0 would greatly simplify the second equation. Then can v be parallel to E? Or must it be perpendicular also?

(If you don't believe in an ansatz, take E = (E, 0, 0) and B = (0, B, 0) and v = (vx,vy,vz). You can always choose your coordinate system that way, but in the end you have to figure out how to describe it without coordinate systems, which can be a bit ugly).

E can't be parallel to v or B' won't vanish, right?

Yes. So since now we know that v must be perpendicular to E and B, we can write v = s E x B, with s a scalar constant. Now you just plug in this v into your equation for B', use some vector analysis and find out s. Then, |v| = |s| |E| |B|.

(BTW: v can have a component parallel to E, but that won't contribute in the equation for B'. So what you calculate above is really the minimum velocity you need to make B' vanish. The question is ambiguous.)

## What is a Lorentz transformation where the electric field vanishes?

A Lorentz transformation is a mathematical equation used in special relativity to describe how measurements of time and space change between two different reference frames. When the electric field vanishes, it means that there is no net electric charge in the reference frame being observed.

## Why is it important to consider a Lorentz transformation where the electric field vanishes?

In special relativity, the laws of physics should be the same in all inertial reference frames. Therefore, understanding how measurements of time and space change when there is no electric field present is crucial for accurately describing physical phenomena.

## How does a Lorentz transformation where the electric field vanishes affect the perception of time and space?

A Lorentz transformation where the electric field vanishes can lead to a phenomenon known as time dilation, where time appears to pass slower for an observer in one reference frame compared to another. It also affects the perception of space, causing length contraction where an object appears shorter in the direction of motion.

## What are some real-life applications of a Lorentz transformation where the electric field vanishes?

One real-life application is in high-speed particle accelerators, where particles are accelerated to close to the speed of light. In this scenario, the electric field may not remain constant, and understanding the Lorentz transformation helps scientists accurately measure and predict the behavior of these particles.

## Are there any limitations to using a Lorentz transformation where the electric field vanishes?

While the Lorentz transformation is a fundamental equation in special relativity, it is only valid for inertial reference frames. In situations involving accelerating objects or strong gravitational fields, more complex equations may be necessary to accurately describe the behavior of time and space.

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