After having consulted numerous physics books, forums and videos, in all cases I found that the electromagnetic wave equation is derived from taking as a premise that the medium is a vacuum. This causes the charge density and current density to be zero in Maxwell's equations, which leads to the electromagnetic wave equation. Unfortunately, the only explanations I have found for making such a premise are:
- "Electromagnetic waves can travel in a vacuum."
- "The electric and magnetic fields of a wave are perpendicular to each other and, in turn, the wave vector is perpendicular to these."
- "Not taking this premise into account causes the electromagnetic wave equation to be inhomogeneous."
Personally, I find these explanations insufficient, since if I were Maxwell trying to derive the electromagnetic wave equation in the 19th century, how would I go about knowing these facts beforehand?
That said, I would love to know what Maxwell realized that allowed him to make that assumption.
Best Answer
Taking the curl of Faraday's law yields $$\nabla \times\big(\nabla \times \mathbf E) = \nabla \times \big(-\frac{\partial \mathbf B}{\partial t}\big)$$ $$\implies -\nabla^2\mathbf E + \nabla(\nabla \cdot \mathbf E) = -\frac{\partial}{\partial t}(\nabla \times \mathbf B) = -\frac{\partial}{\partial t}\left(\mu_0 \mathbf J + \epsilon_0 \mu_0 \frac{\partial \mathbf E}{\partial t}\right)$$ $$ \implies \left(\epsilon_0\mu_0 \frac{\partial^2}{\partial t^2}-\nabla^2\right) \mathbf E = \frac{\nabla \rho}{\epsilon_0} - \mu_0 \frac{\partial \mathbf J}{\partial t}$$ where we've utilized Ampere's law and Gauss' law. We observe that $\mathbf E$ obeys the inhomogeneous wave equation with source $\nabla\rho/\epsilon_0 - \mu_0 \partial \mathbf J/\partial t$. Clearly if $\rho=\mathbf J=0$, then this equation reduces to the homogeneous case, which is a good starting point simply because it's easy to solve; it's also interesting because apparently electromagnetic waves can travel in vacuum, and do so with speed $c \equiv 1/\sqrt{\epsilon_0\mu_0}$.
I'm not sure what you mean by something allowing Maxwell to make an assumption. You can make whatever assumptions you want, under any circumstances. Whether those assumptions yield an accurate model of the phenomena you're trying to understand is a different question, but when faced with a difficult problem it is usually a good idea to solve the simplest cases to gain some intuition for what's going on.