Two free positive charges 4q and q are a distance apart. What charge Q is needed to achieve equilibrium for the entire system and where show it be placed from charge q ?

Explanation

For the system to be in equilibrium, the net force on each charge should be zero.
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Hence, the third charge should be negative and it should be placed near q between (1) and (2)
For equilibrium (3),
`1/(4 pi epsilon_(0)) (qQ)/(x^2)=1/(4 pi epsilon_(0))(Q4q)/((l-x)^(2))`
or `[(l-x)/(x)]^(2) =4 or (l-x)/(x)=2 or x=l/3`
For equilibrium of `(1) 1/(4 pi epsilon_(0))(q4q)/(l^2)=1/(4 pi epsilon_(0))(Qq)/(x^2)`
or `Q=4q(x/l)^(2)=4q(1/3)^(2)=(4q)/(9)`
or `Q=-(4q)/(9)`.