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Can someone please elaborate the following statement from Electricity and Magnetism, by Edward M. Purcell, in an easy to understand way?

the unit vector in $\mathbf F_{21}=\frac{1}{4\pi\varepsilon_0}\frac{q_1q_1}{r_{21}^2} \hat{\mathbf r}_{21}$ shows that the force is parallel to the line joining the two charges. It could not be otherwise unless space itself has some built-in directional property, for with no two point charges alone in empty and isotropic space, no other direction could be singled out.

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  • $\begingroup$ i do not understand what kind of built-in directional property and how does that fit here to show the force is central $\endgroup$
    – Syed Ilyas
    Commented Dec 31, 2016 at 16:14
  • $\begingroup$ When you place the second (test) charge in the field of the first (source) charge, you single out a direction, and it's not that space has some built-in anisotropy. $\endgroup$
    – SRS
    Commented Dec 31, 2016 at 16:29

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The author means space itself is isotropic. Hence for a system of two charges, the only special direction is along the line joining the two charges.

You can prove this by symmetry. If the force is not along the line but makes and angle with it. Then if you rotate space about the line joining the two axis, the force vector will rotate and point to another directions as well. While for two point charges, rotating them about the line joining them should give no change, and therefore this leads to contradiction.

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  • $\begingroup$ Would the downvoter please comment on what being wrong in my answer? $\endgroup$
    – velut luna
    Commented Dec 31, 2016 at 16:33

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