# Physical implication of $\textbf{E}\rightarrow\textbf{B},~~\textbf{B}\rightarrow -\textbf{E}$ invariance of the Maxwell's equations

An interesting observation to consider about the Maxwell's equation is that in absence of the sources, the equations are symmetric under the interchange $$\textbf{E}\rightarrow\textbf{B},~~\textbf{B}\rightarrow -\textbf{E}\tag{1}$$ apart from some numerical factors of $\mu_0$ and $\epsilon_0$. In terms of the electromagnetic field strength tensor $F^{\mu\nu}$ and its dual $\tilde{F}^{\mu\nu}$ the transformation (1) is equivalent to $$F^{\mu\nu}\to \tilde{F}^{\mu\nu},~~\tilde{F}^{\mu\nu}\to -F^{\mu\nu}.\tag{2}$$ It also turns out that if both electric and magnetic sources$^1$, $j^\mu$ and $k^\mu$, are included the duality can still be preserved if Eq.(2) is supplemented with $$j^\mu\to k^\mu,~~k^\mu\to -j^\mu.\tag{3}$$

Does this invariance have any physical significance/consequence in classical electrodynamics itself, in particular, if magnetic sources were there?

$^1$ Inclusion of magnetic sources modify the homogeneous Maxwell's equations to $\partial_\mu\tilde{F}^{\mu\nu}=k^\nu\neq 0$. This implies that the four-potential $A^\mu$ cannot be defined.

Another observation is that if we define $\mathbf{F}:=\mathbf{E}+ic\mathbf{B}$ the duality achieves $\mathbf{F}\mapsto -i\mathbf{F}$, suggesting complex exponential waves will play a natural role in describing electromagnetism.