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Maxwell equations read $$\nabla\cdot \vec E=\rho\tag1$$ $$\nabla\times \vec E = -\frac{\partial \vec B}{\partial t}\tag2$$ $$\nabla\cdot\vec B=0\tag3$$ $$\nabla\times\vec B=\vec j+\frac{\partial\vec E}{\partial t}\tag4$$ For the sake of simplicity, I assume $\vec{j}=0$. Equations (2) and (3) form a linear first order system $$D_x {\bf X}(t,x) = \partial_t ... 2 The Maxwell's equations are the basics of EM phenomenon. Whatever be the fields you select, they shouldn't violate these fundamental 4 equations. Suppose we are provided a problem to find the electric and magnetic fields of an EM wave or a charge, or whatever be it. As you said, we have now a four component problem. But the degree of freedom is not 4 as each ... 2 I think the answer is simply: "Yes". What you should keep in mind is energy conservation: As long as there are no sources, the total energy of the electromagnetic field is conserved. But then what, in free space, would cause the initial change in the electric or magnetic field to get the oscillations going? A source, which is possibly localized ... 1 The quality of the vacuum at LHC is pretty good but particles are constantly accelerated/bent by strong electromagnetic fields along the circumference of the accelerator. Thus the situation is not the one felt by a free particle. So conceptually, in your diagram, you can simply replace the photon coming from the nucleus by a photon from the electromagnetic ... 1 Lets look at the 4 equations in ED,$$\nabla\cdot \vec E=\rho\tag1\nabla\times \vec E = -\frac{\partial \vec B}{\partial t}\tag2\nabla\cdot\vec B=0\tag3\nabla\times\vec B=\vec j+\frac{\partial\vec E}{\partial t}\tag4$$which can ofcourse be written in a more compacted form,$$\partial_\mu F^{\mu\nu}=j^\nu \tag5 The $(2)$ and the $(3)$ ...