Three charges `+Q_(1),+Q_(2)` and q are placed on a straight line such that q is somewhere in between `+Q_(1)`and `Q_(2)`. If this system of charges is in equilibrium what should be the magnitude and sign of charge q?

Options

A. `(Q_(1)Q_(2))/((sqrt(Q_(1))+sqrt(Q_(2))))`, positive
B. `(Q_(1)Q_(2))/(2)`, positive
C. `(Q_(1)Q_(2))/((sqrt(Q_(1))+sqrt(Q_(2)))^(2))`, negative
D. `(Q_(1)+Q_(2))/(2)`, negative

Answer

C

Explanation

Force on `Q_(2)` is zero (q should be negative)

`(kQ_(1)Q_(2))/(R^(2))= (kqQ_(2))/(x^(2))` or `(x)/(R) = sqrt((q)/(Q_(1)))`
Force on q is zero:
`(kQ_(1)q)/((R-x)^(2)) = (kqQ_(2))/(x^(2))`
or `(R-x)/(x) = sqrt((Q_(1))/(Q_(2)))`
`rArr (R)/(x) = (sqrt(Q_(1))+sqrt(Q_(2)))/(sqrt(Q_(3)))`or`(sqrt(Q_(1)))/(sqrt(q))=(sqrt(Q_(1))+sqrt(Q_(3)))/(sqrt(Q_(2)))` or `q = (Q_(1)Q_(2))/((sqrt(Q_(1))+sqrt(Q_(2)))^(2))`