Transformation of Electromagnetic Fields

We are now in a position to discuss the transformation behavior of electric and magnetic fields under a change of the coordinate system, i.e., their transformation properties under Lorentz transformations. Since the electric and magnetic fields do not represent Lorentz 4-vectors, we have to analyze the transformation property of the field strength tensor Fi instead. 

Similar calculations yield for the other components of the electric field 

The transformation behavior for the components of the magnetic field could similarly be derived by exploiting Eq. (3.171). A more elegant and direct approach, however, is based on the duality transformation given by Eq. (3.174), which immediately yields 

Since our constraint of a relative motion in a -direction (i.e., v = vex) was arbitrary and only imposed in order to simplify the calculations above, we may find the transformation properties of electric and magnetic fields for the general motion of IS relative to IS as described by the velocity v to read 

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