# How to minimise sum potential of the system?

Given $$m$$ number of immobile electric charges at some points of the plane. We want to place $$n$$ number of further charged particles (not including the fixed charges) along a circle centered at the origin, in such a way that the sum potential of the system is minimized.

Which point should I take to calculate the sum potential of the system? Should I take the origin (the center point of $$n$$ charges)?

If yes, is it correct to calculate the sum potential of the system according to the formula?

$$\sum V = \sum\limits_{i=1}^m k\frac{Q_i}{r}+n k\frac{Q_n}{r}$$

For example (let's suppose distance in cm)

$$\sum V = k\frac{Q_1}{0.03}+k\frac{Q_2}{0.045}+4 k\frac{Q_n}{0.04} = k\frac{Q_1}{0.03}+k\frac{Q_2}{0.045}+100k Q_n$$

Is it correct, in order to minimize the sum of potential, $$N_j$$ point should be placed as far as possible from the origin?

• Are all the charges of the same sign? And are we allowed to choose the radius of the circle or is that a fixed given quantity? Apr 19, 2020 at 11:29
• @SuperfastJellyfish, signs are not given. As they are not given, I think they should be the same sign. Yes, we are allowed to choose the radius of the circle and it is not given. I should put them in the way, sum potential should be minimized Apr 19, 2020 at 11:40

Under the given conditions, since you are adding extra positive terms the minimum will occur when the extra terms are minimum. Lowest value the extra terms can take is zero for radius $$R\to\infty$$