Two pith balls carrying equal charges are suspended from a common point by strings of equal length; the equilibrium separation between them is r. Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now becomes

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(2) r/(√[3]2). Let the free length of each string be L and the separation s. For the small angles involved, tan θ≈(s/2)/L, and the horizontal balance on either ball is (Fₑ)/(mg)=tan θ⇒(kq²)/(s² mg)=s/(2L) s³=(2kq²L)/(mg), so s∝ L^1/3 Clamping the strings at half the height halves the free length, so (s')/r=((L/2)/L)^1/3=1/(2^1/3) s'=r/(√[3]2)≈0.794 r The cube root is the whole point: the charge is unchanged, so the balance (kq²)/(s²)=(mgs)/(2L) ties s³ to L, not s to L. Beware the derivation that reuses the old separation r in the new geometry - it gives r/√2, and the compilation prints exactly that in place of the cube root.

Two pith balls carrying equal charges are suspended from a common point by strings of equal length; the equilibrium separation between them is r. Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now becomes

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