Two point charges `+q_(1) and +q_(2)` are located at two points with position vectors `vecr_(1) and vecr_(2)`. Find a negative charge `q_(3)` and the position vector `vecr_(3)` of the point at which it has to be placed for the force acting on each of the three charges to be equal to zero.

Explanation

For the equilibrium of `q_(3)`

`(1)/(4pi in_(0)) {(q_(2)q_(3)(bar r_(2)-barr_(3)))/(|barr_(2)-barr_(3)|^(3)) +(q_(1)q_(3)(barr_(1)-barr_(3)))/(|barr_(1)-barr_(3)|^(3))}=0`
but `(barr_(2) -barr_(3))/(barr_(2)-barr_(3))=-(barr_(1)-barr_(3))/(|barr_(1)-barr_(3)|) or (q_(2))/(|barr_(2)-barr_(3)|^(2))=(q_(1))/(|barr_(1)-barr_(3)|^(2))`
or `sqrt(q_(2)) (barr_(1) -barr_(3))=sqrt(q_(1)(barr_(3) -barr_(2))`
`rArr barr_(3)=(sqrt(q_(2))barr_(1)+sqrt(q_(1))barr_(2))/(sqrt(q_(1))+sqrt(q_(2)))`
For the equilibrium of `q_(1)`
`(1)/(4pi in_(0)) {(q_(3)(barr_(1)-barr_(3)))/(|barr_(1)-barr_(3)|^(3)) +(q_(2)(barr_(2)-barr_(1)))/(|barr_(2)-barr_(1)|^(3))}=0` or `q_(3)=(q_(2)|barr_(1)-barr_(3)|^(2))/(|barr_(2)-barr_(1)|^(2))`
Substituting the value of `r_(3)`,
we get `q_(3)= (-q_(1), q_(2))/( (sqrt(q_(1)) +sqrt(q_(2)))^(2))`