Two fixed point charges `+4e` and `+e` units are separated by a distance a. Where should a third point charge be placed for it to be in equilibrium?

Explanation

Suppose the charges are placed as shown in fig. As the charge +e exerts repulsive force F on charge +4e, so for the equilibrium of charge + 4e, the charge –q must exert attraction F' on +4e. This requires the charge q to be negative. For equilibrium of charge +4e,
`F=F'`
`1/(4piepsilon_(0))(4exxe)/(a^2)=1/(4piepsilon_(0)).(4exxq)/(x^(2))`

or `q=(ex^2)/(a^(2))`
For equilibrium of charge –q,
`F_1` between +4e and –q
`F_2` between + e and – q
`1/(4piepsilon_(0))(4exxe)/(x^2)=1/(4piepsilon_(0)).(4exxq)/((a-x^(2)))`
or `x^2=4(a-x)^2" " :.x=2a//3`
Hence `q=(ex^2)/(a^2)=e/a^2.(4a^2)/(9)=(4e)/9`