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Question Regarding Electromagnetic Fields in Special Relativity

  1. May 5, 2009 #1
    According to the http://en.wikipedia.org/wiki/Biot-Savart_Law" [Broken], the equation for the magnetic field around a charged particle moving with constant velocity is

    \mathbf{B} = \frac{1}{c^2} \mathbf{v} \times \mathbf{E}

    But then, http://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field" [Broken], the relativistic description for the magnetic field, where B and E are the nonrelativistic magnetic and electric fields is

    \mathbf{B} ' = \gamma ( \mathbf{B} - \frac{1}{c^2} \mathbf{v} \times \mathbf{E})- \frac{\gamma - 1}{v^2} ( \mathbf{B} \cdot \mathbf{v} ) \mathbf{v}

    But this would mean that B' is always 0. Am I misunderstanding something? Is one of these equations the wrong one to use?

    Would the B in the relativistic equation be zero to begin with?
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. May 5, 2009 #2
    In the first equation v is velocity of the particle, while in the second one the same letter is used to mean relative velocity of reference frames. If these two velocities are equal then B' = 0 because the particle is at rest in the '-reference frame.
  4. May 5, 2009 #3


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    Read carefully, the Wiki article on BS says
    "Note that the law is only approximate," for the equation B= qvXr/r^3.
    To put it more clearly, that equation is wrong.
    The correct equations for E and B are the last two equations in that section.
    They do include B=vXE, but E is the complicated field, not the NR Coulomb field.
    The second equation you give is for a Lorentz transformation of the fields, and refers to different things than your first equation.
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