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 0.107 N when their center-to-center separation is 46.6 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 0.0321 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

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 = 0.107 N, and
F2 = kq3^2/r^2 = 0.107 N, where 2q3 = q1+q2 due to conservation of charge.
Solving for q3,
q3 = sqrt(0.107r^2/k) = 7.8658E-7 C
Then solving for q1 we have
F1 = -kq1(2*7.8658E-7-q1)/r^2 = 0.107 which yields a quadratic
-q1^2 + 2*7.8658E-7q1 + 0.107r^2/k = 0 resulting in
q1 = 2.37293E-6 C and q2 = -7.99772E-7 C, which sum to 2*7.8658E-7 C.

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