Electric field in different frames of reference

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tamir
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I study electromagnetism and I got to the chapter about special relativity, in this chapter my professor (since we are not using the electromagnetic tensor in this course) used a specific case to show that the electric field parallel to the velocity of a frame of reference stay the same in both the original frame and the moving frame.

However when I look at the given situation of a point charge +q moving in the x-axis with velocity v, relative to a frame called S, and I calculate the electric field in both the S frame and in the point charge frame (S'), I get different values for the electric field in each frame.

Say the particle cross the origin of the S frame at t=0 at the S frame, and at t'=0 in his frame, and we want to calculate the field he generates at (x,0,0) (coordinates of S), what I have done is:
In the S frame I used the regular equation of the electric field of a point charge E=q/x^2 and in the S' frame I also used this equation E'=q/x'^2 and used lorentz transformation and got E'=q/(γ*x)^2.
Other students told me I should have got q/(γ*x)^2 in both frames and from some reason I can't calculate the electric field of the charge in the frame he moving in using E=q/r^2, but I don't know why.
 
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