Classical solutions for the Lorentz-violating and $CPT$-even term of the standard model extension

In this work, we calculate the classical solutions for the electrodynamics stemming from the Lorentz-violating (LV) and $CPT$-even term of the standard model extension. Static and stationary solutions for pointlike and extended charges are obtained from the wave equations by means of the Green method. A dipolar expansion is written for the field strengths. It is explicitly shown that charge and current generate LV first order effects for the magnetic and electric fields, respectively. Using the magnetic field generated by a macroscopic $1C$ charged sphere, we establish a stringent bound for the LV parameter: $\kappa \le {10}^{-16}$.