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.)
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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