On the Divergenceless Property of the Magnetic Induction Field

Maxwell's equations beautifully describe the electromagnetic fields properties. In what follows we will be interested in giving a new perspective to divergence-free Maxwell’s equations regarding the magnetic induction field: . To this end we will consider some physical aspects of a system consisting of massive nonrelativistic charged particles, as sources of an electromagnetic field (e.m.) propagating in free space. In particular the link between conservation of total momentum and divergence-free condition for the magnetic induction field will be deeply investigated. This study presents a new context in which the necessary condition for the divergence-free property of the magnetic induction field in the whole space, known as solenoidality condition, directly comes from the conservation of total momentum for the system, that is, sources and field. This work, in general, leads to results that leave some open questions on the existence, or at least the observability, of magnetic monopoles, theoretically plausible only under suitable symmetry assumptions as we will show. 1. Introduction The elegant description of the electromagnetic field given by Maxwell’s equations is a universally accepted milestone in physics. Many text books describe these equations in detail, together with their relative applications [1]. In what follows we will essentially discuss the Maxwell equation regarding the divergence-free property of the magnetic induction field, in order to give a new interpretation of it. We will consider a system formed by massive, nonrelativistic charged particles as moving sources of the electromagnetic field propagating in a homogeneous, isotropic, and linear space. The main idea concerning the link between total momentum [2] and solenoidality of field has been suggested by the structure of Lorentz’s equation in which magnetic induction acts perpendicularly to the particle’s velocities such that no work variation occurs. In an isolated system, total momentum is a constant of motion [3, 4]; starting from this invariance we will deduce the necessary condition for the solenoidality of . This work leaves some open questions on the existence, or at least the observability, of magnetic monopoles; the reader interested in these topics can refer to the extensive bibliography in the literature for details [5–7]. Anyway this paper will discuss special symmetries in which the assumption of nonsolenoidality for the magnetic induction field and the consequently existence of magnetic monopoles could subsist consistently with the conservation of the total momentum.