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 0.090 N when their center to center separation is 53.00 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 0.030 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

let the initial charges be q and q ' separation of the charges r = 53 cm =0.53 m attractive force F = 0.09 N Kqq' / r^ 2 = 0.09 qq ' = 0.09 r^ 2 / K where K = coulomb's constant = 8.99 * 10 ^ 9 Nm^ 2/ C^ 2 plug the values we get qq ' = -2.81* 10 ^ -12 ----( 1) Since one is positive and other is negative charge,because they attaract each other charge on each sphere after they touch Q = ( q + q' ) / 2 repulsive force F ' = KQ^ 2/ r^ 2 Q ^ 2 = F' r^ 2 / K where F ' = 0.03 N = 9.37 * 10 ^ -13 Q = 9.68 * 10 ^-7 q + q' = 1.936* 10 ^ -6 ----( 2) since Q = (q + q' )/ 2 q - q ' = sqrt {( q + q') ^ 2 -4qq'} = 3.87* 10 ^-6 C ---( 3) from eq( 2) and ( 3) , q = [eq( 2) + eq( 3) ] / 2 = 2.903* 10 ^ -6 C and q ' = -0.967* 10^-6 C

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