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Transformation of Fields

The electromagnetic field tensor transforms according to the standard rule

$\displaystyle F^{\,\mu'\nu'} = F^{\,\mu\nu} \,p_\mu^{\,\mu'} \,p_{\nu}^{\,\nu'}.$ (1806)

This easily yields the celebrated rules for transforming electromagnetic fields:

$\displaystyle E_\parallel'$ $\displaystyle = E_\parallel,$ (1807)
$\displaystyle B_\parallel'$ $\displaystyle = B_\parallel,$ (1808)
$\displaystyle {\bf E}_\perp'$ $\displaystyle = \gamma\,({\bf E}_\perp +{\bf v}\times{\bf B}),$ (1809)
$\displaystyle {\bf B}_\perp'$ $\displaystyle = \gamma\,({\bf B}_\perp-{\bf v}\times{\bf E}/c^2),$ (1810)

where $ {\bf v}$ is the relative velocity between the primed and unprimed frames, and the perpendicular and parallel directions are, respectively, perpendicular and parallel to $ {\bf v}$ .

At this stage, we may conveniently note two important invariants of the electromagnetic field. They are

$\displaystyle \frac{1}{2}\, F_{\mu\nu} \,F^{\,\mu\nu} = c^{\,2} \,B^{\,2} - E^{\,2},$ (1811)

and

$\displaystyle \frac{1}{4} \,G_{\mu\nu} \,F^{\,\mu\nu} = c\,{\bf E} \cdot {\bf B}.$ (1812)

The first of these quantities is a proper-scalar, and the second a pseudo-scalar.


next up previous
Next: Potential Due to a Up: Relativity and Electromagnetism Previous: Dual Electromagnetic Field Tensor
Richard Fitzpatrick 2014-06-27