Charged particle in a B field, tensor notation

[fluidistic]

Homework Statement

A charged particle of charge q with arbitrary velocity $\vec v_0$ enters a region with a constant $\vec B_0$ field.
1)Write down the covariant equations of motion for the particle, without considering the radiation of the particle.
2)Find $x^\mu (\tau)$
3)Find the Lorentz transformation such that the motion of the particle is restricted to a plane.
4)Calculate the dipole term of the fields and the average radiated EM power by the particle.

Homework Equations

$\tau=t\gamma$

The Attempt at a Solution

1)$\frac{dp^\alpha}{d\tau}=qU_\beta F^{\alpha \beta}$ would be my reply.
Where $p^\alpha$ is the four-momentum, $U_\beta$ is the four-velocity and $F^{\alpha \beta}$ is the electromagnetic tensor.
2)I solved the equation found in 1) by writing the tensors under matrix form. I got that $x^\alpha=(ct,x^1(t),x^2(t),0)$ where $x^1(t)=\frac{mv_{0x}}{qB_0}\sin \left ( \frac{qB_0 t}{m} \right ) + x^1_0$ and $x^2(t)=\frac{mv_{0x}}{qB_0}\cos \left ( \frac{qB_0 t}{m} \right ) + x^2_0$. Now I just realize that they asked in terms of $\tau$ and not t. Well I get $x^1(\tau)=\frac{mv_{0x}}{qB_0\sqrt{\gamma}}\sin \left ( \frac{qB_0\sqrt{\gamma}\tau}{m} \right )$.
3)I'm out of ideas on this one. I know I can express the tensor of the Lorentz transformation as a matrix (I guess this way: https://en.wikipedia.org/wiki/Lorentz_transformation#Matrix_forms) but I'm not sure what they're asking for when they ask for the motion to be in a plane. Isn't the motion already in the x-y plane in my case? I picked $\vec B_0$ as $B_0\hat z$ originally.

 

[Orodruin]

but I'm not sure what they're asking for when they ask for the motion to be in a plane. Isn't the motion already in the x-y plane in my case?

Yea, your motion is in a plane, but it is not the most general solution to the equations of motion. In trying to solve 2, you solved the special case 3.

 

[fluidistic]

Yea, your motion is in a plane, but it is not the most general solution to the equations of motion. In trying to solve 2, you solved the special case 3.

I see. Is it because I chose the magnetic field to be oriented in the z direction? I would have thought that I'd need a boost in a particular direction to solve 3), not picking a convenient orientation.

 

[Orodruin]

I see. Is it because I chose the magnetic field to be oriented in the z direction?

It is the combination of selecting the B-field in the z-direction and assuming that there is no velocity component in the z-direction.

 

[fluidistic]

It is the combination of selecting the B-field in the z-direction and assuming that there is no velocity component in the z-direction.

Well then I miserably failed, even at solving part 3 since there's no way I can pick the velocity to vanish in the z direction (unless I apply a boost maybe?). I'll try to solve the problem in a correct way. Which means, I believe, that I can pick the orientation for the B field to be in the z direction but I can't assume that there's no speed in the z direction.

 

[fluidistic]

Ok I've redone part 2. I had made some mistakes previously I think.
I get $$x^\mu (\tau) = (c\gamma \tau , \gamma ^2 v_{0x}\frac{B_0q}{m}\sin \left ( \frac{m\tau }{q\gamma B_0} \right ) +x_0 , -\frac{mv_{0x}}{qB_0}\cos \left ( \frac{m \tau}{q\gamma B_0} \right ) +y_0, z_0 )$$.
The fact that I get different amplitudes in front of the sine and cosine really bothers me. This would make an elliptical trajectory in the x-y plane projection instead of circular path. Am I doing things wrong?

 

[Orodruin]

The easiest way to check is to insert your result into the original differential equation ...