There are two identical positively charged conducting sphere

There are two identical, positively charged conducting spheres fixed in space. The spheres are 30.8 cm apart (center to center) and repel each other with an electrostatic force of F1 = 0.0765 N. Then, a thin conducting wire connects the spheres, redistributing the charge on each sphere. When the wire is removed the spheres still repel but with a force of F2 = 0.115 N. Using this information, find the initial charge on each sphere, q1 and q2 if initially q1<q2.

Solution

let the initial charges being q1 and q2.

then force of repulsion = kq1q2/0.308^2 = 0.0765 N

so, q1q2= 8.063 x 10^-13

when wire connects the sphere, charge on each sphere becomes equal to (q1+q2)/2

then force of repulsion = k (q1+q2)^2/(4*0.308^2) = 0.115

so, (q1+q2)^2 = 48.486 x 10^-13

so, q1+q2 = 2.201 x 10^-6 C .......................... (1)

so, (q1-q2)^2 = [(q1+q2)^2-4q1q2]=(48.486 x 10^-13 - 4 x 8.063 x 10^-13)=16.234 x 10^-13

so, q2-q1 = 1.274 x 10^-6 C .......................... (2)

adding (1) and (2) and dividing by 2

q2= 1.7375 x 10^-6 C

and q1 =0.4635 x 10^-6 C