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Magnetic Fields with Relativistic Motion of Electron

  1. Jun 7, 2013 #1
    1. The problem statement, all variables and given/known data
    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?

    2. Relevant 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

    3. 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.
     
  2. jcsd
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