The electric field of an electromagnetic wave propagating in a...

An electromagnetic wave is produced by an accelerating or oscillating charge, not by a charge moving with uniform velocity. Thus, the source charge must be oscillating.

The given electric field equation $\vec{E} = E_0 \sin(kz - \omega t) \hat{i}$ indicates that the wave propagates along the z-axis and the electric field oscillates along the x-axis (due to the $\hat{i}$ direction).

Since the electric field of the radiated wave is parallel to the direction of acceleration of the source charge, the charge must be oscillating along the x-axis.

The angular frequency of the wave is $\omega$. The linear frequency of the wave, and hence the frequency of the oscillating source charge, is $f = \dfrac{\omega}{2\pi}$.

Answer: Oscillating along the x-axis with a frequency of $\dfrac{\omega}{2\pi}$