Coulomb's Law — Question 7

Question 7

A total charge and force measurement determine an unordered pair of charges

Use $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$; treat the specified charges as points in vacuum. Two small conducting spheres, far enough apart to use a point approximation, have total charge $+10.0$ nC. At separation .300 m they repel with force $2.0976667\times10^{-6}$ N (rounded from an exact model value). They are then brought into a symmetric contact arrangement, disconnected, and returned to the same separation. The spheres are identical.

1. Infer the two original charges, retaining the two possible assignments to labeled spheres.

2. Find their final charges and the ratio of final to initial force. Explain why sharing increases the force for these same-sign charges.

Question 7 - Solution

A total charge and force measurement determine an unordered pair of charges

(a) Method and calculation

$$q_A+q_B=10\,\mathrm{nC},\quad q_Aq_B=Fr^2/k\approx21\,\mathrm{nC^2}.$$$$z^2-10z+21=0\Rightarrow z=3\text{ or }7,\quad\{q_A,q_B\}=\{3,7\}\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$q_A\prime=q_B\prime=5\,\mathrm{nC},\quad F\prime/F=25/21\approx1.19048.$$$$q_Aq_B=\frac{(q_A+q_B)^2-(q_A-q_B)^2}{4}\le\frac{(q_A+q_B)^2}{4}.$$Equal sharing maximizes the product at fixed sum; the initial measurements do not identify which sphere originally carried 3 nC.

Physical check

The rounded measured force supports approximately 3 and 7 nC; it does not justify more digits than the input precision.