# Relativistic Lorentz Force with constant, perpendiuclar, and uniform E and B fields

I am trying to solve the following problem from the Fundamentals of Plasma Physics by Bittencourt (Problem 2.7):

Analyze the motion of a relativistic charged particle in the presence of crossed electric $\bf{E}$ and magnetic $\bf{B}$ fields that are constant in time and uniform in space. What coordinate transform must be made in order to transform away the transversal electric field? Derive equations for the velocity and trajectory of the charged particle.

I am able to write down the equation for the relativistic Lorenz force in the following form:

$\frac{d}{dt}(m\gamma\vec{v})= m\gamma \frac{d\vec{v}}{dt} + q\frac{\vec{v}}{c^2}(\vec{v}\cdot\vec{E})= q(\vec{E}+\vec{v}\times\vec{B})$

But I am not sure how to proceed solving for the velocities and the trajectories or how to "transform away the transversal electric field". Is it necessary to first "transform away the transversal electric field" to solve for the trajectories/velocities?

Any help is greatly appreciated, thank you.

• Do we know if one of the fields is bigger than the other? If you fix your $\vec B$ field strength and make the electric field weaker and weaker then eventually there is a point where the electric field is nonzero in your original frame, but is zero in another frame. In that other frame the particle sees just a magnetic field so the motion is fairly simple in that frame. To know if you are small enough, compare the sign of the scalar invariant (your pseudoscalar invariant is zero because of orthogonality) to the sign of the scalar invariant for a pure magnetic field. – Timaeus May 26 '15 at 3:21
• I have posted the problem as it appears in its entirety in the book I am using; their is no mention of a relation between the magnitudes of the $\vec{E}$ and $\vec{B}$ fields. Could you please clarify some of the terms you are using? First, what do you mean by "compare the sign of the scalar invariant to the scalar invariant for a pure magnetic field"? I think the scalar and pseudo scalar terms that you are talking about are respectively $E^2 - c^2B^2$ and $\vec{E}\cdot\vec{B}$ is that correct? If so, I still do not understand their relevance here. – Loonuh May 26 '15 at 3:33

When the pseudoscalar invariant $\vec E \cdot \vec B$ is zero, we have three cases.
If $E^2<c^2B^2$ then you can switch to a frame moving with speed $E/B$ in a direction mutually orthogonal to $\vec E$ and $\vec B$ where there is no electric field in the new frame. Solve in the new frame. Then bring it back to the original frame.
If $E^2>c^2B^2$ then you can switch to a frame moving with speed $c^2B/E$ in a direction mutually orthogonal to your $\vec E$ and $\vec B$ where there is no magnetic field in the new frame. Solve in the new frame. Then bring it back to the original frame.
If $E^2=c^2B^2$ then I'm not familiar with any advantage to choosing any particular frame to compute in since I don't notice the electromagnetic field aligning in any particular way with any particular frame.
• With respect to the question then, it seems that in order to transform away the electric field, it must the case that $E^2 < c^2B^2$? Also, what is the actual "transform" that needs to be implemented? I have a sense that I should be using a "Lorentz transform" here, but I'm not extremely skilled with Special Relativity, could you maybe point me to a specific reference that I could use to understand this better? Thank you. – Loonuh May 26 '15 at 16:41