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 0.117 N when their center- to-center separation is 55.0 cm. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres have a net positive charge and repel each other with an electrostatic force of 0.0370 N. What was the initial negative charge on one of the spheres, and what was the initial positive charge on the other? (Hint: Use charge conservation and solve for one of the initial charges. You will end up with a quadratic equation. The solutions give you the positive and negative charges. Include the sign of the value in your answer.) initial negative charge initial positive charge

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 = 0.0117 N, and
F2 = kq3^2/r^2 = 0.0370 N, where 2q3 = q1+q2 due to conservation of charge.
Solving for q3,
q3 = sqrt(0.0370r^2/k) = 3.900E-7 C
Then solving for q1 we have
F1 = -kq1(2*3.900E-7-q1)/r^2 = 0.177 which yields a quadratic
-q1^2 + 2*3.900E-7q1 + 0.117r^2/k = 0 resulting in

-q1^2 + 7.8*10^-7E-7q1 + 0.117r^2/9*10^9 = 0

-q1^2 + 7.8*10^-7E-7q1 + 1.521*10^-12= 0

q1 = +1.6834E-6 C

and q2 = -9.0348E-7 C,

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