Transformations of the Electromagnetic Field in Different Frames of Reference and the Applications

The transformation of the electromagnetic fields in different frames of reference (wether inertial reference frames or non-inertial reference frames) is the problem frequently met during the electromagnetic measurements in space and the relative analysis. For example, to draw the values of the electromagnetic fields in spacecraft comoving reference frame from the electromagnetic fields measured in the spacecraft rotational reference frame with a reasonable accuracy. Another example is the calculation of the charge density based on the four-point electrostatic field observations of MMS; the present analysis is not very satisfactory and there is still no rigid evaluation on the method applied. However, it is not easy to find a plain and rigid evaluation on the transformation formulas used. In this research, a systematic theoretic investigation has been performed, the universal formulas for the transformation are given and further applied to two actual situations successfully. For space plasmas, the relative velocities of the structures are generally very low and always much less than the speed of light in vacuum, so that the Galillia transformations are applicable. In this study, the Galillia transformations of the electromagnetic fields, the electric potential and magnetic vector potential, and the charge density and current density in different reference frames (wether inertial reference frames or non-inertial reference frames) have been presented and the respective errors are given. The results can find wide applications in space physics. At first, the general formula for the rotational potential of the planets are obtained. Secondly, by using the yielded theoretical results, a strict verification on the deduction of the charge density based on MMS electrostatic fields measurements has been made. It is found that, the Poisson equation is valid because the Coulomb gauge can be used for low-speed motions, and it is enough to draw the charge density from the MMS electrostatic fields measurements with a first order relative error.