Two spheres? (Electric)?

Question

Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of 0.0911 N when their center-to-center separation is 43.3 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 0.0297 N. Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge in coulombs on one of them and (b) the positive charge in coulombs on the other?

Answer 1

This can be treated like a point-charge problem, since we are measuring over distances greater than the sphere radii. Then

F1 = -kq1q2/r^2 = 0.0911 N, and

F2 = kq3^2/r^2 = 0.0297 N, where 2q3 = q1+q2 due to conservation of charge.

Solving for q3,

q3 = sqrt(0.0297r^2/k) = 7.8658E-7 C

Then solving for q1 we have

F1 = -kq1(2*7.8658E-7-q1)/r^2 = 0.0911 which yields a quadratic

-q1^2 + 2*7.8658E-7q1 + 0.0911r^2/k = 0 resulting in

q1 = 2.37293E-6 C and q2 = -7.99772E-7 C, which sum to 2*7.8658E-7 C.