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On the infinities in the electric potential of unbounded charge distributions

In every beginner electricity and magnetism course, a good teacher usually insists that if one has an infinite charged plate or an infinity charged wire, one must first calculate the electric field to find the potential difference between two space points by making a linear integral of the field. One does it this way because Coulomb’s potential, i.e. the expression which allows for a direct calculation of the potential without first calculating the electric field, should not be used since these charge distributions are not bounded in space. Due to the unbounded nature of these distributions, Coulomb’s potential diverges at all points of space in the two previously mentioned cases. However, in this work, we are going to demonstrate that in fact one can use Coulomb’s potential to find the correct results by properly removing the infinities that appear in the potential of these distributions. This is possible because electric potential, like potential energy, is only defined up to an additive constant, which in the mentioned cases can be chosen as an infinite constant. One can show that the removal of these infinities is a simple example of the renormalization techniques commonly used in quantum field theory.