General relation for the vector magnetic field of a circular current loop: a closer look

This paper presents a general theory for the fields generated by a circular current loop and compares it with existing theories. The existing, general, closed solution for the vector magnetic field may be expressed in a number of seemingly different but equivalent forms. These relations offer alternative closed-form solutions that may find various applications, including the characterization of Helmholtz coils. The paper provides alternative closed forms in both spherical and cylindrical coordinate systems. It employs Gauss´s magnetic law to show that the alternative closed form is self-consistent and correct and also shows agreement with well-known solutions. Finally, it develops a new (or not readily found) tabulated mathematical identity.