what must the value of the third be?

Question

Three point charges which initially are infinitely far apart are placed at the corners of an equilateral triangle with sides d. two of the pt charges are identical and have charge q. if zero net work is required to place the three charges at the corners of the triangle,

Answer 1

Let the third charge to be kq, where k is a unit-less parameter to be found. Also let r to be the side length of the equilateral triangle. So the net work done to place the three charges at the corners of the triangle is the total potential energy of the system:
PE = (q^2 + 2*kq^2)/(4*pi*r*Epsilon_0)
k must be -1/2 for PE = 0.
So the third charge is -q/2.