Deriving Coulomb's law from symmetry and torsion-balance measurements

Step 1 of 4: Force along the joining line

Intuition

Two charged balls in empty space 'know' about each other only through the segment joining them — with no other landmarks, the force can only push along that segment.

\vec{F}_{12} = f(q_1, q_2, r)\,\hat{r}_{12}

Analysis

For two point charges, the only direction defined by the problem is the line joining them. The isotropy of empty space forbids the force from 'leaning' toward either side perpendicular to that line — nothing distinguishes the two sides. Thus $\vec{F}_{12}$ must be collinear with $\hat{r}_{12}$ ; only the magnitude $f$ remains to be found.