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If I have an electric field that its direction is parallel to the direction of the wave propagation, it will not satisfy Gauss's law for vacuum. However we can say it satisfies Gauss's law for non-zero charge density (mathematically). For this case:

  1. Does that mean this electric field is practically feasible for an EM wave (for non-zero charge densities)?

  2. Will the other 3 laws of Maxwell be violated?

Furthermore, If the direction of the wave propagation is perpendicular to the direction of the electric field, does this guarantee that the field vector satisfy all Maxwell's equation?

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  • $\begingroup$ For an electromagnetic field in vacuum the electric field has vanishing divergence outside the source region and there it is transverse (to see this consider its Fourier transform). Maxwell's equations will be satisfied. As to your last question, note that the field is a solution of Maxwell's equations. If you have a concrete expression for your wave and it is not a solution of Maxwell's equations then it is not a proper electromagnetic wave. $\endgroup$
    – Urgje
    Commented Feb 2, 2015 at 9:14

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By definition, an electromagnetic wave is a solution to Maxwell's equations in vacuum. The electric field of a EM wave solution is always perpendicular to the direction of propagation. Let me denote this electric field by $\vec{E}_{EM}$. If $\vec{v}$ is the velocity of the wave, then we must have $\vec{E}_{EM} \cdot \vec{v} = 0 $.

However, Maxwell's equations are linear equations and in particular allow for superposition of solutions. By cleverly choosing sources, it seems entirely likely that we can construct an electric field generated by the source $\vec{E}_{source}$ in such a way that the total electric field $\vec{E}_T = \vec{E}_{EM} + \vec{E}_{source}$ is parallel to $\vec{v}$.

Further, due to the superposition princpiple, if $\vec{E}_{EM}$ and $\vec{E}_{source}$ satisfy Maxwell's equations then so does $\vec{E}_T$.

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An electromagnetic wave satisfies $\mathbf E = \mathbf v \times \mathbf B$ and therefore the electric and magnetic fields are always perpendicular to the direction of motion in vacuum.

Any electric and magnetic field must satisfy Maxwell's equations, for they won't be physically allowable otherwise.

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