Three point charges q, 2q and 8q are to be placed on a .9cm long straight line. Find the positions where the charges should be placed such that the potential energy of this system is minimum. In this situation, what is the electric field at the charge q due to the other two charges?

Explanation

The maximum contribution may come from the
charge 8q forming pairs with others. To reduce its effect,
it should be placed at a corner and the smallest charge
q in the middle. This arragement shown in (figure)
ensures that the charges in the strogest pair
2q, 8q are at the largest seperation.
The potential energy is
`U = (q^2)/(4piepsilon_0) [ (2)/(x) + (16)/(9cm) + (8)/(9cm -x)]`
. This will be minimum if
`A =(2)/(x) +(8)/(9cm-x)` is minimum.
for this, `(dA)/(dx)= - (2)/(x^2) + (8)/(9cm-x)^2=0`
or 9 cm-x = 2x or, x=3 cm.
The electric field at the position of charge q is
`(q)/(4piepsilon_0) ((2)/(x^2) -(8)/(9 cm - x)^2).`
= 0.