Why we use $E$ field for the EM-Wave and not the electric displacement field or electric induction? Why we use E field for the EM-Wave when is propatating on vacuum and not the electric displacement field or electric induction?
 A: The E field is the physical field.  The D field is a mathematical convenience used to simplify some equations. As one comment said, it is the E field, not the D field, that exerts force.
A: In the simplest case, the wave equation admits, in addition to sinusoidal and cosinusoidal solutions, a complex exponential solution. The electric displacement easily satisfies the exponential solution $\vec{D}=\hat{D} \ e^{i\left(\omega t-kx\right)}$which in vacuum implies the existence of a displacement vector with two components, one transverse equal to $\varepsilon_o \vec{E}$ and another longitudinal equal to the polarization $\vec{P}$ of the vacuum. Maxwell's equations admit the polarization of the vacuum when an electromagnetic wave propagates in that medium, whatever the frequency of the wave, that is, for any frequency greater than zero. This does not imply movement of charge in space, neither in translation nor in rotation. It simply varies the value of the charges of both signs while they remain stationary in space. In the simplest case, the variation is sinusoidal and the phenomenon produces what we call an electromagnetic wave, because this charge variation is virtually equivalent to charge moving in the direction of propagation. This constitutes what we call displacement current in vacuum, corresponding to the virtual movement mentioned.
