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In inertial frame $\mathcal{O}$, a region of space-time is filled with constant electric field $\vec{E}$ and magnetic field $\vec{B}$. Another inertial frame $\mathcal{O}'$ has 3-velocity $\vec{V}$ relative to $\mathcal{O}$. What is the electromagnetic field $\left(\vec{E}', \vec{B}'\right)$ measured in $\mathcal{O}'$? Express the result in terms of $\vec{E}$, $\vec{B}$, $\vec{V}$, dot product ($\cdot$) and cross product ($\times$).

This is to get the general formula for the boost transformation of the electromagnetic fields. I know the general form of the Lorentz boost transformation. So, obviously the solution for this problem seems to be applying this boost transformation to the electromagnetic field tensor $F^{uv}$. That is, for the boost transformation $\Lambda^u_v$, calculate $F'^{ab}=\Lambda^a_u \Lambda^b_vF^{uv}$. But this seems like a tremendous amount of work... Is there any more efficient solution than this? Could anyone suggest me?

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  • $\begingroup$ Look at how $A^a$ transforms. $\endgroup$
    – J.G.
    Commented Apr 1, 2018 at 17:27
  • $\begingroup$ Then I also must see how $\partial_a$ transforms. Right? $\endgroup$
    – Keith
    Commented Apr 1, 2018 at 17:28
  • $\begingroup$ You're making life too hard for yourself. Forget calculus altogether at the start: just write $\phi',\,\mathbf{A}'$ in terms of their unprimed counterparts. $\endgroup$
    – J.G.
    Commented Apr 1, 2018 at 17:34
  • $\begingroup$ Use w.l.o.g.w.c.a.t. (without loss of generality, we can assume that). For example, assume that the velocity is in the $z$-direction, derive some formulae, then generalize back to dots and crosses based on the patterns. The answer is also in Jackson's electromagnetism textbook. $\endgroup$ Commented Apr 1, 2018 at 18:06
  • $\begingroup$ In my answer here : Is it a typo in David Tong's derivation of spin-orbit interaction? for the configuration of two inertial systems as in Figure-01 the space-time Lorentz transformation is given by equations (03). The Lorentz transformation of the electromagnetic field is given by equations (04). I would post in an answer the proof of this transformation of the electromagnetic field only if a moderator untags the question as "homework-and-exercises". $\endgroup$
    – Frobenius
    Commented Jan 28, 2020 at 8:51

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Yes of course. There are three different methods at least

  • using transformation of the electromagnetic field tensor $F^{\mu\nu}$ (which is TOO LONG!!!)
  • using the transformation of 4-vector potential $\mathbf A^\mu$
  • using the Lorentz force $\mathbf{F}=q \mathbf{E}+q \mathbf{v}\times\mathbf{B}$
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