Measuring Unknown Charges with a Coulomb Torsion Balance

Problem Statement

A physics lab experiment uses a Coulomb torsion balance to determine the charges on two small metal spheres A and B. The spheres are identical conducting spheres of the same size.

From prior qualitative tests (triboelectric series, etc.), it is known that sphere A carries positive charge $+Q_A$+QA​ ($Q_A > 0$QA​>0) and sphere B carries negative charge $-Q_B$−QB​ ($Q_B > 0$QB​>0), with $Q_A > Q_B$QA​>QB​.

1. Spheres A and B are placed with their centers a distance $r$r apart. The twist angle of the torsion wire shows an attractive force of magnitude $F_1$F1​ acting between them.

2. Using an insulating rod, spheres A and B are brought into contact once. After charge redistribution is complete, the spheres are separated again to center-to-center distance $r$r.

3. Now a repulsive force of magnitude $F_2$F2​ is observed between the two spheres.

No charge leaks to the surroundings. Each sphere is small enough relative to $r$r to be treated as a point charge.

From these observations, find the initial charge magnitudes $Q_A$QA​ and $Q_B$QB​.

Constraints

• Coulomb's law constant: $k = 9.0 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$k=9.0×109 N⋅m2/C2
• Center-to-center distance: $r = 3.0 \times 10^{-2} \text{ m}$r=3.0×10−2 m
• Attractive force before contact: $F_1 = 2.4 \times 10^{-4} \text{ N}$F1​=2.4×10−4 N
• Repulsive force after contact: $F_2 = 1.0 \times 10^{-5} \text{ N}$F2​=1.0×10−5 N

Input Format

The values $Q_A$QA​ and $Q_B$QB​ can each be written as $x \times 10^{-9} \text{ C}$x×10−9 C and $y \times 10^{-9} \text{ C}$y×10−9 C respectively (where $x, y$x,y are positive integers).

Compute $10x + y$10x+y and give the answer as a positive integer.