Threee 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 equlibirium what should be the magnitude and sign of charge q?

Options

A. `(Q_(1)Q_(2))/((sqrt(Q_(1))+sqrt(Q_(2)))^(2))`, positive
B. `(Q_(1)+Q_(2))/(2)`, positive
C. `(Q_(1)Q_(2))/((sqrt(Q_(1))+sqrtQ_(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))`
`(R-x)/(x)=(sqrt(Q_(1)))/(sqrt(Q_(2)))`
or, `(R )/(x)=(sqrt(Q_(1))+sqrt(Q_(2)))/(sqrt(Q_(2)))` or `(sqrt(Q_1))/(sqrt(q))=(sqrt(Q_1)+sqrt(Q_2))/(sqrt(Q_2))`
or `q=(Q_(1)Q_(2))/((sqrt(Q_1)+sqrt(Q_2))^(2))`.