Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of -0.9537 n when separated by 50 cm, center-to-center. 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 0.0756 n.
what were the initial charges on the spheres?

The initial charges on the spheres are   and

Explanation:

Electrostatic Force

Two charges q1 and q2 separated a distance d exert a force on each other which magnitude is computed by the known Coulomb's formula

We are given the distance between two unknown charges d=50 cm = 0.5 m and the attractive force of -0.9537 N. This means both charges are opposite signs.

With these conditions we set the equation

Rearranging

Solving for q1.q2

The second part of the problem states the spheres are later connected by a conducting wire which is removed, and then, the spheres repel each other with an electrostatic force of 0.0756 N.

The conducting wire makes the charges on both spheres to balance, i.e. free electrons of the negative charge pass to the positive charge and they finally have the same charge:

Using this second condition:

Solving for q2

Replacing in [1]

Rearranging, we have a second-degree equation for q1.  

Solving, we have two possible solutions

Which yields to two solutions for q2

Regardless of their order, the initial charges on the spheres are and

the first, second, and fifth options