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

first , let the charges be q1 and -q2

so F = 9*10^9 q1q2 / d^2

=> 0.123 = 9*10^9 q1q2 / 0.686^2

hence q1q2 = 6.43 * 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 , 0.0483 = 9*10^9 * Q^2 / 0.686^2

=> Q1 = Q2 = 1.59 uC

so Q1 + Q2 = 2Q = 3.18 uC = q1 + q2 => q2 = 3.18uC - q1

from eq 1

q1q2 = 6.43 * 10^-12

=> q1 (3.18 uC - q1) = 6.43 * 10^-12

3.18 * 10^-6 q1 - q1^2 = 6.43 * 10^-12

=> q1^2 - 3.18 x 10^-6 q1 + 6.43 * 10^-12 = 0

q1 = [3.18 * 10^-6 +/- sqrt [ (3.18 * 10^-6)^2 - 4(6.43 * 10^-12) ] ] / 2

q1 = 1.787 uC

q2 = -1.39 uC

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