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The unit vector $\hat{r}_{21}$ shows that the force is parallel to the line joing the charges. It could be otherwise unless space itself has some built-in directional property. . . This can be predicted by logical arguments from symmetry. - Berkely Physics Course Vol. 2 by Edward M Purcell.

Now, the question seems to be totally nonsense; but I do actually want to know why is the force acting parallel to the line joining charges. Or what led to assume(??) that the force is acting in that direction; it could be anywhere acting. How can I use symmetry to prove this? Plz help.

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  • $\begingroup$ We're gonna need a bit more of context here. For exemple a drawing of your charge distribution. $\endgroup$ – Michaël Ughetto Mar 18 '15 at 13:33
  • $\begingroup$ What do you mean by "cause"? We define the electric field such that $\vec E = q\vec F$, so by definition the force is parallel to the field line joining a positive and negative charge. $\endgroup$ – ACuriousMind Mar 18 '15 at 13:37
  • $\begingroup$ @ACuriousMind: Actually, sir, don't take it too literally. I only want to know is it a consequence of experimental verification or assumed(to - be - correct) empirical law. That's all:) $\endgroup$ – user36790 Mar 18 '15 at 13:48
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Start from the assumption that space is isotropic (independent of direction). Under that assumption, if you rotate your imagined empty-universe-with-two-charges by any angle around the line joining the two charges, then you will wind up with exactly the same physical system you started with. In particular, the charge configuration is unchanged.

Now, if the electric force on one of the charges has a component perpendicular to the line, then that component will be affected by the rotation, and thus the force acting on the charge will be different after the rotation than before it. But this means that two different forces ($\vec{F}_\text{after}$ and $\vec{F}_\text{before}$) are produced by the exact same physical system. There is a uniqueness theorem in electromagnetism that says this is impossible: any given configuration of charges produces exactly one electromagnetic field. So either space is not isotropic, or the electric force on each charge has no component perpendicular to the line.

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  • $\begingroup$ +1. Method of Contradiction! But sir, can you please elaborate to me how the "isotropy-assumption" leads to the same system after any type of rotation about the line joining the charges? $\endgroup$ – user36790 Mar 18 '15 at 16:19
  • $\begingroup$ Can you elaborate on what's not clear about it to you? That's practically the definition of isotropy. $\endgroup$ – David Z Mar 18 '15 at 18:35
  • $\begingroup$ How is direction independence related to the status of the system after its rotation? Angular displacement is not a vector quantity! Moreover, if it is assigning direction independence, does it mean that the perpendicular component at any direction will be same?? $\endgroup$ – user36790 Mar 18 '15 at 18:59
  • $\begingroup$ Indeed angular displacement is not a vector quantity (it's a pseudovector), but that doesn't matter. The only physical quantity in the problem is force, so you only need to consider how the force acts under a rotation. The argument in my post is how you conclude that the perpendicular components in all directions must be the same. $\endgroup$ – David Z Mar 19 '15 at 4:42

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