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 0.123 N when their center-to-center separation is 68.6 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 0.0483 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

We can solve it by treating it as a point charge problem since the distances are greater than the sphere radii.

So,

F1 = -kq1q2/r2 = 0.123 N,

F2 = kq32/r2 = 0.0483 N,

where q32 = q1+q2 due to conservation of charge.

Solving for q3,

q3 = sqrt(0.0483*0.6862/k) =1.589*10-6 C

Then solving for q1 we have

F1 = -kq1(2*1.589*10-6 *-q1)/r2 = 0.123

which yields a quadratic

-q12 + 2*1.589*10-6 q1 + 0.123r2/k = 0

-q12 + 3.178*10-6 q1 + 6.43*10-12 = 0

resulting in

q1 = -1.4*10-6 C and q2 = 4.58*10-6 C, which sum to 3.18*10-6 C.

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