Electric field in different frames of reference

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In the discussion on electric fields in different frames of reference, a student studying electromagnetism highlights a discrepancy in electric field calculations for a moving point charge. The student notes that while the electric field parallel to the velocity remains constant, their calculations yield different values in the stationary frame (S) and the moving frame (S'). Specifically, they apply the electric field equation for a point charge in both frames but encounter confusion regarding the Lorentz transformation. Other students suggest that the electric field should be consistent across both frames, but the student struggles to reconcile this with their results. The conversation emphasizes the complexities of applying special relativity to electromagnetism.
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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