Lorentz boost to obtain parallel E and B fields?

  • #1

Homework Statement


Suppose given an electric field [itex]\vec{E}[/itex] and a magnetic field [itex]\vec{B}[/itex] in some inertial frame. Determine the conditions under which there exists a Lorentz transformation to another inertial frame in which [itex]\vec{E} || \vec{B}[/itex]

Homework Equations


If we give a Lorentz boost along [itex]x_1[/itex]-direction, then in the boosted frame, electric and magnetic fields are given by
[tex]E_1' = E_1\\
E_2' = \gamma (E_2 - \beta B_3)\\
E_3' = \gamma (E_3 + \beta B_2)[/tex]
And similar for components of B fields.

The Attempt at a Solution


I started with a frame in which the fields are parallel and see what kind of fields I can obtain after the transformation. The case where the boost is along the direction of E//B fields is trivial. Then I consider the case where I boost in the direction perpendicular to the E//B fields. By the equations I listed I find that I can produce E and B fields with some angle depending on [itex]\beta[/itex]. But I am not seeing how I can go further from here. Am I in the right direction? Or should I try some other approach?
 

Answers and Replies

  • #2
phyzguy
Science Advisor
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One way to approach problems like these is to construct invariants - i.e quantities that are the same in all frames. For example, given a four-vector A, the quantity [itex] A^\mu A_\mu[/itex] has the same value in all frames. It turns out there are two invariants that can be constructed from the electromagnetic field tensor [itex] F^{\mu \nu}[/itex], and hence from E and B. Try to discover what these are - this will help answer the question.
 
  • #3
One way to approach problems like these is to construct invariants - i.e quantities that are the same in all frames. For example, given a four-vector A, the quantity [itex] A^\mu A_\mu[/itex] has the same value in all frames. It turns out there are two invariants that can be constructed from the electromagnetic field tensor [itex] F^{\mu \nu}[/itex], and hence from E and B. Try to discover what these are - this will help answer the question.
Thanks for the hint. I realize that [itex]F_{\rho \sigma}\tilde{F}^{\rho\sigma} = -4 \vec{B}\cdot \vec{E}[/itex], which is a scalar. That means if E//B in one frame, [itex] \vec{E}\cdot \vec{B} \neq 0[/itex] in all frames. Thus I just need to find the right Lorentz boost for systems like that. I will try and see what I can find.
 

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