Past exam of the physics 2 paper of the University of Oxford / 2013 resit / 7 / i / Solution

Put $q_1$ at $x=-L$, $q_2$ at $x=+L$, and $q_3$ at $x=fL$, with $-1<f<1$. For a nonzero test charge, the force vanishes when the electric field vanishes:For equal-sign nonzero $q_1,q_2$, take the positive square root $s=\sqrt{q_2/q_1}$. The distances are positive, so $1-f=s(1+f)$ andFor $s>1$ the zero moves toward the weaker charge $q_1$; equal charges give $f=0$.