I'm taking E & M II this semester, and one question got me thinking. We know the formulation of the four Maxwell's equations, and that's okay so far. But in the absence of sources, they take the form:
$$\nabla \cdot \mathbf{E}= 0 \\
\nabla \cdot \mathbf{B}= 0 \\
\nabla \times \mathbf{E} = -\dfrac{\partial\mathbf{B}}{\partial t}\\
\nabla \times \mathbf{B} = \mu_{0}\epsilon_{0}\dfrac{\partial\mathbf{E}}{\partial t}$$
The electric field is caused by the presence of charged particles, which if it varies, causes the existence of a magnetic field. But if there aren't any sources, how can the Maxwell's equations have nontrivial solutions? Wouldn't the electric and magnetic field always be zero?
Best Answer
Think about a simpler problem. You have the following differential equation: $$f'=0$$ Does this mean that $f$ has to be zero? No. It's a possible solution but not the only one. You need to provide the value of $f$ somewhere.
As for Maxwell's equations, zero fields are a possible solution, but not the only one. If the fields are non-zero somewhere (boundary conditions), then a solution with zero fields may no longer be acceptable.
For example, if you are in the dead of space (no charge and no current), you can still have, for example, wave-like solutions describing an electromagnetic wave coming from far away and crossing that space.