A number of questions have previously been asked about electrostatic energy for a system of conductors. But I am still confused about this topic.
Suppose we have an ideal capacitor consisting of two spheres both of radius a. One sphere has the positive charge $Q_A$ = Q and its center is located at point A. The other sphere has the negative charge $Q_B$ = -Q and its center is located at point B. Whatever the size of the radius a, the electrostatic energy W for this system is given by $W =\frac{QV}{2}$ where $V = ɸ_A – ɸ_B$, the potential difference between the conductors. $ɸ_A$ is larger than $ɸ_B$ so W is positive. Now we let a tend to $0$ until we have two point charges Q and -Q. We conclude that W for the two point charges is positive.
But that is wrong. Having two point charges $Q_A$ = Q and $Q_B$ = -Q we can still use $W =\frac{QV}{2}$ if we define $ɸ_A$ as the potential at point A due to $Q_B$ with $Q_A$ absent and $ɸ_B$ as the potential at point B due to $Q_A$ with $Q_B$ absent. Then we get $W = -\frac{1}{4\pi\epsilon_0}\frac{Q^2}{r}$ where r is the distance between $Q_A$ and $Q_B$. This formula is correct as it follows immediately from the definition of W. So W is actually negative for a system consisting of two point charges Q and -Q.
Now I wonder, what went wrong in the first argument?