Two identical conducting spheres, fixed in place, attract each other with an ele

Question

Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of 0.132 N when their center-to-center separation is 49.0 cm. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of 0.0381 N. Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge on one of them and (b) the positive charge on the other? (Assume the negative charge has smaller magnitude.)

Explanation / Answer

k Q^2 / .49^2 = .0381      Q = 1.01 * 10E-6
k Q1 Q2 / .49^2 = .132     Q1 Q2 = -3.52 * 10E-12      originally attractive
Q = Q1 - q
Q = Q2 + q       where q is charge transferred
2 Q = Q1 + Q2 = 2.02 * 10E-6
Q1 - Q2 = 2 q
Q^2 = Q1 Q2 + q (Q1 - Q2) - q^2      multiplying above equations
Q^2 = Q1 Q2 + q^2     substituting for (Q1 - Q2)
q^2 = Q^2 - Q1 Q2 = ((1.02 + 3.52) * 10E-12
q = 2.13 * 10E-6     charge transferred
Q1 = 3.14 * 10E-6
Q2 = -1.12 * 10E-6      using equations from above
Check:
F = k Q1 Q2 / .49^2 = -9 * 10E9 * 3.14 * 1.12 * 10E-12 = -.132

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