Let denote the coulomb constant. Let denote the distance between the two point charges. In this question, neither and depend on the value of .
By Coulomb's Law, the magnitude of electrostatic force between and would be:
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Find the first and second derivative of with respect to . (Note that .)
First derivative:
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Second derivative:
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The value of the coulomb constant is greater than . Thus, the value of the second derivative of with respect to would be negative for all real . would be convex over all .
By the convexity of with respect to , there would be a unique that globally maximizes . The first derivative of with respect to should be for that particular . In other words:
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In other words, the force between the two point charges would be maximized when the charge is evenly split: