Problem 3.4

Two positive charges $q_{1}$ and $q_{2}$ are located at the points with radius vectors $\mathbf{r}_ {\mathbf{1}}$ and $\mathrm{r}_ {2} .$ Find a negative charge $q_{3}$ and a radius vector $\mathrm{r}_{3}$ of the point at which it has to be placed for the force acting on each of the three charges to be equal to zero.

q_{3}=-\frac{q_{1} q_{2}}{\left(\sqrt{q_{1}}+\sqrt{q_{2}}\right)^{2}}, \quad \mathbf{r}_{3}=\frac{\mathbf{r}_{1} \sqrt{q_{2}}+\mathbf{r}_{2} \sqrt{q_{1}}}{\sqrt{q_{1}}+\sqrt{q_{2}}}

Constant Electric Field in a Vacuum