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The electromagnetic field tensor transforms according to the standard
rule
![\begin{displaymath}
F^{\mu'\nu'} = F^{\mu\nu} p_\mu^{\mu'} p_{\nu}^{\nu'}.
\end{displaymath}](img3092.png) |
(1511) |
This easily yields the celebrated rules for transforming electromagnetic
fields:
![$\displaystyle E_\parallel'$](img3093.png) |
![$\textstyle =$](img41.png) |
![$\displaystyle E_\parallel,$](img3094.png) |
(1512) |
![$\displaystyle B_\parallel'$](img3095.png) |
![$\textstyle =$](img41.png) |
![$\displaystyle B_\parallel,$](img3096.png) |
(1513) |
![$\displaystyle {\bf E}_\perp'$](img3097.png) |
![$\textstyle =$](img41.png) |
![$\displaystyle \gamma ({\bf E}_\perp +{\bf v}\times{\bf B}),$](img3098.png) |
(1514) |
![$\displaystyle {\bf B}_\perp'$](img3099.png) |
![$\textstyle =$](img41.png) |
![$\displaystyle \gamma ({\bf B}_\perp-{\bf v}\times{\bf E}/c^2),$](img3100.png) |
(1515) |
where
is the relative velocity between the primed and unprimed
frames, and the perpendicular and parallel directions are, respectively,
perpendicular and parallel to
.
At this stage, we may conveniently note two important invariants of the
electromagnetic field. They are
![\begin{displaymath}
\frac{1}{2} F_{\mu\nu} F^{\mu\nu} = c^2 B^2 - E^2,
\end{displaymath}](img3101.png) |
(1516) |
and
![\begin{displaymath}
\frac{1}{4} G_{\mu\nu} F^{\mu\nu} = c {\bf E}\!\cdot \! {\bf B}.
\end{displaymath}](img3102.png) |
(1517) |
The first of these quantities is a proper-scalar, and the second
a pseudo-scalar.
Next: Potential due to a
Up: Relativity and electromagnetism
Previous: The dual electromagnetic field
Richard Fitzpatrick
2006-02-02