Electric Potential due to a System of Charges

What this figure shows. A field point P surrounded by several discrete point charges $q_1,q_2,\ldots,q_n$ scattered at various positions, each joined to P by a line labelled with its own distance $r_1,r_2,\ldots,r_n$. No force or field arrows are drawn (unlike the corresponding superposition figure for fields) -- only the charges and their individual distances to P -- visually emphasising that computing the NET potential at P needs nothing more than these scalar distances and the individual charge mag …

Worked out. Charges $q_1=5\times10^{-8}$ C and $q_2=-3\times10^{-8}$ C are 16 cm apart on the x-axis, with $q_1$ at the origin. Setting the sum of their potentials to zero at a point P, $x$ from the origin: $\dfrac{1}{4\pi\epsilon_0}\left(\dfrac{q_1}{x}+\dfrac{q_2}{0.16-x}\right)=0$ for P between the charges gives, after simplifying, $x=0.10$ m -- a point 10 cm from $q_1$ (and hence 6 cm from $q_2$) where the larger-but-farther positive contribution exactly cancels the smaller-but-closer negative one. A second zero-potential point also exists OUTSIDE the segment, beyond $q_2$ on the far side (along the extended line, where $q_2$'s closer negative contribution can still be cancelled by $q_1$'s larger charge from farther away): solving the corresponding equation for that configuration gives $x=0.40$ m $=40$ cm from the origin -- illustrating that a syste …