Can someone help me with this Physics problem?

Question

A large insulating sphere of radius a carries a total positive charge Q with a uniform volume charge density. A very thin tunnel is drilled through a diameter of the sphere and two small spheres with charge q are placed in the tunnel. These spheres are represented by the blue dots in the figure. They find equilibrium positions at a distance of r on either side of the center of the sphere. Your research director has had great success with this scheme.

(a)Determine the specific value of r at which equilibrium exists. (Use any variable or symbol stated above as necessary.)

Answer 1

I would appear that q must be negative if Q is positive.

The magnitude of the E field at a distance r from the center is k Qhat / r²

where Qhat is the volume charge inside of radius r . Qhat = (r/a)³ Q

The force on one of the q charges due to the volume charge is F_v = k Qhat |q| / r²

This force is opposed by the force of repulsion between the two q charges F_d = k q²/( 4 r² )

Equating F_v and F_d , and using the expression for Qhat leads to

q²/ 4 = Q |q| (r / a)³ so

r = a ( |q| /( 4 Q) ) ^1/3