An ab initio derivation of the electromagnetic fields of a point charge   in arbitrary motion
  Electromagnetic fields of an accelerated charge are derived from the first principles using Coulomb's law and the relativistic transformations. The electric and magnetic fields are derived first for an instantaneous rest frame of the accelerated charge, without making explicit use of Gauss's law, an approach different from that available in the literature. Thereafter we calculate the electromagnetic fields for an accelerated charge having a non-relativistic motion. The expressions for these fields, supposedly accurate only to a first order in velocity $\beta$, surprisingly yield all terms exactly for the acceleration fields, only missing a factor $1-\beta^2$ in the velocity fields. The derivation explicitly shows the genesis of various terms in the field expressions, when expressed with respect to the time retarded position of the charge. A straightforward transformation from the instantaneous rest frame, using relativistic Doppler factors, yields expressions of the electromagnetic fields for the charge moving with an arbitrary velocity. The field expressions are derived without using Li\'{e}nard-Wiechert potentials, thereby avoiding evaluation of any spatial or temporal derivatives of these potentials at the retarded time. 
Ashok K. Singal
physics.class-ph astro-ph.CO astro-ph.HE