# Magnetic Fields with Relativistic Motion of Electron

1. Jun 7, 2013

### clarinethero

1. The problem statement, all variables and given/known data
There is a magnetic field given by the equation $\overrightarrow{B}=B_{0} \hat{x} \sin\left(2\pi z/L\right)$.

If there is a $10^7 eV$ electron going in the $\hat{z}$ direction (moving at a relativistic velocity), a mag. field strength of 0.1T, and a mag. period of 0.01m, what are the field values?

2. Relevant equations

Eq. 1 $E_{k}=\gamma m c^2$

Eq. 2 $\gamma = \dfrac{1}{\sqrt{1-v^2/c^2}}$

3. The attempt at a solution

By using Eq. 1, I can find $\gamma$ from the mass of the electron and given energy. This gets me $\gamma = 19.57$. If I don't consider $\gamma$ my results would be highly inaccurate, as this electron is traveling at relativistic speeds.

I can then find the velocity of the electron using Eq. 2 now that I know $\gamma$. This gets me $v = 2.99\times 10^8 m/s$.

The magnetic field can then be found using the provided mag. field equation. I know that $B_{0} = 0.1T$. I believe $L= 0.01m$. However, how do I find out what $z$ is? This is something I do not understand.

The value of the magnetic field taking into account the relativistic speed, $\overrightarrow{B}'$, can be found (maybe? I'm basing this off the Wiki) by $\overrightarrow{B}'=\gamma \overrightarrow{B}$ now that I know $\overrightarrow{B}$ from before.

The value of the electric field produced from the relativistic speed of the electron can be found from $\overrightarrow{E}'=\gamma v \overrightarrow{B_{x}} \hat{y}$. I'm basing this, once more, off of the Wiki page.

I'm not sure if my equations are correct, but my main issue here is how to find what $z$ is. Any help would be appreciated. I'm not looking for the answer - just a suggestion of where to go.