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## 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?