Answer:
[tex]q / q_{1} = 2[/tex], assuming that [tex]q_{1}[/tex] and [tex](q - q_{1})[/tex] are point charges.
Explanation:
Let [tex]k[/tex] denote the coulomb constant. Let [tex]r[/tex] denote the distance between the two point charges. In this question, neither [tex]k[/tex] and [tex]r[/tex] depend on the value of [tex]q_{1}[/tex].
By Coulomb's Law, the magnitude of electrostatic force between [tex]q_{1}[/tex] and [tex](q - q_{1})[/tex] would be:
[tex]\begin{aligned}F &= \frac{k\, q_{1}\, (q - q_{1})}{r^{2}} \\ &= \frac{k}{r^{2}}\, (q\, q_{1} - {q_{1}}^{2})\end{aligned}[/tex].
Find the first and second derivative of [tex]F[/tex] with respect to [tex]q_{1}[/tex]. (Note that [tex]0 < q_{1} < q[/tex].)
First derivative:
[tex]\begin{aligned}\frac{d}{d q_{1}}[F] &= \frac{d}{d q_{1}} \left[\frac{k}{r^{2}}\, (q\, q_{1} - {q_{1}}^{2})\right] \\ &= \frac{k}{r^{2}}\, \left[\frac{d}{d q_{1}} [q\, q_{1}] - \frac{d}{d q_{1}}[{q_{1}}^{2}]\right]\\ &= \frac{k}{r^{2}}\, (q - 2\, q_{1})\end{aligned}[/tex].
Second derivative:
[tex]\begin{aligned}\frac{d^{2}}{{d q_{1}}^{2}}[F] &= \frac{d}{d q_{1}} \left[\frac{k}{r^{2}}\, (q - 2\, q_{1})\right] \\ &= \frac{(-2)\, k}{r^{2}}\end{aligned}[/tex].
The value of the coulomb constant [tex]k[/tex] is greater than [tex]0[/tex]. Thus, the value of the second derivative of [tex]F[/tex] with respect to [tex]q_{1}[/tex] would be negative for all real [tex]r[/tex]. [tex]F\![/tex] would be convex over all [tex]q_{1}[/tex].
By the convexity of [tex]\! F[/tex] with respect to [tex]\! q_{1} \![/tex], there would be a unique [tex]q_{1}[/tex] that globally maximizes [tex]F[/tex]. The first derivative of [tex]F\![/tex] with respect to [tex]q_{1}\![/tex] should be [tex]0[/tex] for that particular [tex]\! q_{1}[/tex]. In other words:
[tex]\displaystyle \frac{k}{r^{2}}\, (q - 2\, q_{1}) = 0[/tex].
[tex]2\, q_{1} = q[/tex].
[tex]q_{1} = q / 2[/tex].
In other words, the force between the two point charges would be maximized when the charge is evenly split:
[tex]\begin{aligned} \frac{q}{q_{1}} &= \frac{q}{q / 2} = 2\end{aligned}[/tex].
