Two identical conducting spheres, fixed in place, attract each other with a forc

Question

Two identical conducting spheres, fixed in place, attract each other with a force of 0.100 N when their center to center separation is 43.00 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 0.043 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.)

Initial negative charge _________(C):
Initial positive charge _________(C):

Explanation / Answer

first , let the charges be q1 and -q2

so F = 9x10^9 q1q2 / d^2

=> 0.12 = 9x10^9 q1q2 / 0.41^2

hence q1q2 = 2.24 x 10^-12 ........ (1)

after removing the wire let the charges be Q1 and Q2

since they were connected by a conductor their potential differences are equal , and because they're identical then Q1 = Q2 = Q

so , 0.033 = 9x10^9 x Q^2 / 0.41^2

=> Q1 = Q2 = 0.785 uC

so Q1 + Q2 = 2Q = 1.57 uC = q1 + q2 => q2 = 1.57uC - q1

from eq 1:

q1q2 = 2.24 x 10^-12

=> q1 (1.57uC - q1) = 2.24 x 10^-12

1.57 x 10^-6 q1 - q1^2 = 2.24 x 10^-12

=> q1^2 - 1.57 x 10^-6 q1 + 2.24 x 10^-12 = 0

q1 = [1.57 x 10^-6 +/- sqrt [ (1.57 x 10^-6)^2 - 4(2.24 x 10^-12) ] ] / 2

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