Determine Radius of Zero Potential Circle

To find the radius $R$ of the zero potential circle, we set the total electric potential $V(x,y)$ equal to zero for a general point $(x,y)$ in the $xy$-plane:

$V(x,y) = \frac{1}{4\pi\varepsilon_0} \left( \frac{-Q}{\sqrt{x^2 + y^2}} + \frac{Q/\sqrt{3}}{\sqrt{(x-2)^2 + y^2}} \right) = 0$

Equating the potential contributions gives:

$\sqrt{3}\sqrt{(x-2)^2 + y^2} = \sqrt{x^2 + y^2}$

Squaring both sides and simplifying the equation yields:

$3\left(x^2 - 4x + 4 + y^2\right) = x^2 + y^2 \implies 2x^2 - 12x + 2y^2 + 12 = 0$

Dividing by $2$ and completing the square for $x$:

$(x - 3)^2 + y^2 = 3$

This represents a circle centered at $(3,0)$ with radius $R = \sqrt{3}\text{ m} \approx 1.73\text{ m}$.