Magnetic Fields with Relativistic Motion of Electron

[clarinethero]

Homework Statement

There is a magnetic field given by the equation [itex]\overrightarrow{B}=B_{0} \hat{x} \sin\left(2\pi z/L\right)[/itex].

If there is a [itex] 10^7 eV [/itex] electron going in the [itex] \hat{z} [/itex] 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?

Homework Equations

Eq. 1 [itex] E_{k}=\gamma m c^2 [/itex]

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

The equations on this page: http://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity

The Attempt at a Solution

By using Eq. 1, I can find [itex] \gamma [/itex] from the mass of the electron and given energy. This gets me [itex] \gamma = 19.57 [/itex]. If I don't consider [itex] \gamma [/itex] 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 [itex] \gamma [/itex]. This gets me [itex] v = 2.99\times 10^8 m/s [/itex].

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

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

The value of the electric field produced from the relativistic speed of the electron can be found from [itex] \overrightarrow{E}'=\gamma v \overrightarrow{B_{x}} \hat{y} [/itex]. 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 [itex] z [/itex] is. Any help would be appreciated. I'm not looking for the answer - just a suggestion of where to go.

 

[anathema]

Hi there,

Thank you for sharing your attempt at solving this problem. It seems like you have a good understanding of the equations and concepts involved. To find the value of z, you can use the fact that the electron is traveling in the z-direction and has a period of 0.01m. This means that after traveling a distance of 0.01m, the electron will complete one full cycle in the magnetic field. Therefore, you can set z=0.01m in the given magnetic field equation and solve for B. This will give you the value of the magnetic field at that specific point in space.

Regarding your approach for finding the values of B' and E', it seems correct. Just make sure to use the correct values for B and v in the equations. Also, keep in mind that the electric field will be in the y-direction, perpendicular to both the magnetic field and the direction of motion of the electron.

I hope this helps. Good luck with your calculations!