For an electric charge oscillating sinusoidally along the x-axis, the resulting electric and magnetic fields will propagate as an electromagnetic wave. Here’s the breakdown of the direction of the electric field at the observer’s location on the $+z$-axis:

1. Charge Motion and Field Generation:

   • Since the charge oscillates along the x-axis, the electric field generated at any point due to the oscillation will oscillate in the plane perpendicular to the direction of wave propagation.
   • In this case, the electromagnetic wave propagates along the z-axis toward the observer.

2. Electric Field Direction:

   • For an observer on the $+z$-axis, the electric field will oscillate in a direction perpendicular to the direction of wave propagation (z-axis) and to the direction of the charge’s oscillation (x-axis).
   • This leaves the y-axis as the only option, meaning the electric field at the observer’s location will oscillate along the y-axis.

3. Resulting Orientation:

   • The electric field $\mathbf{E}$ will oscillate along the $y$-axis, and the magnetic field $\mathbf{B}$ will oscillate in a direction perpendicular to both $\mathbf{E}$ and the propagation direction $z$-axis (i.e., along the x-axis).

Summary

Thus, the electric field at the observer's location on the $+z$-axis will oscillate in the y direction.

Would you like more details on electromagnetic wave propagation principles or have other questions?

1. What is the relation between the direction of the magnetic field and the electric field in wave propagation?
2. How does the oscillation frequency of the charge affect the wavelength observed at the +z axis?
3. Why does the electric field oscillate perpendicular to the direction of wave propagation?
4. How does the amplitude of the electric field depend on the distance from the oscillating charge?
5. What would change if the observer was not on the +z axis but at an arbitrary point in space?

Tip: In electromagnetics, remember that the electric and magnetic fields in a wave are always perpendicular to each other and the direction of propagation.