Examination of the affect of Coulombs Law

Coulombs Law is used to estimate the distance of a 4th charged particle from the origin of a co-ordinate system so that no net force is experienced by one of the particles

In general we use the equation for the Coulomb force between 2 charges and separated by distance as below

The factor we will denote by to keep the analysis simple going forward
The total force expereiced by charge due to charges is thus given by (1) below
(1)

(2)
When
(3)
But charges so (3) reduces to
(4)
(5)
Dividing (5) by throughout
(6)
But so and
Thus

Thus (6) becomes

Which has solutions for given by

Only valid solution is for to be positive in the context of the diagram so the solution is

If charges and move closer to the axis but maintain their symmetry then the force between these charges and charge increases. You can see that then the distance from these charges to charge decreases as decreases the distance separating these charges and goes down as decreases

Thus the force due to charge must increase to compensate to keep the net force on charge zero. Thus charge needs to be brought towards charge to compensate so needs to decrease

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