Magnetic Monopoles in Classical Electrodynamics

Abstract

Though in the theoretical interpretation of the electromagnetic phenomena of nature the electric field $\mathbf {E}$ and the magnetic field $\mathbf {B}$ play specular roles in some respects, they exhibit substantial differences in others. A mirror symmetry between these fields is, for instance, evident in Maxwell’s equations in empty space (20.1)–(20.4), which govern the propagation of the electromagnetic waves. These equations involve, in fact, the fields $\mathbf {E}$ and $\mathbf {B}$ on the same footing, except for a minus sign in Faraday’s law of induction (20.3). Similarly, in the Poynting vector (2.147), which quantifies the electromagnetic energy flux, the interchange of $\mathbf {E}$ and $\mathbf {B}$ produces just a minus sign. Furthermore, these fields contribute in exactly the same way to the electromagnetic energy density (2.133). In maximum contrast to these mirror properties, in the Lorentz equation

$$ {d \mathbf {p}\over dt}=e\left( \mathbf {E}+{\mathbf {v}\over c}\times \mathbf {B}\right) $$

the fields $\mathbf {E}$ and $\mathbf {B}$ play completely different roles. In particular, the effects of the magnetic field are suppressed by a relativistic factor v / c with respect to those of the electric field. However, the most significant distinction between these fields emerges in the presence of non-vanishing sources. In this case we have, in fact, the contrasting Gauss’s laws

$$ \varvec{\nabla }\cdot \mathbf {E}= \rho ,\quad \quad \varvec{\nabla }\cdot \mathbf {B}=0, $$

according to which a (static) charge distribution generates an electric field, but no magnetic one. In other words, conventional electrodynamics hosts electric charges, but no magnetic ones.