Two charges `2mu c` and `1 muc`  are placed at a distance of 10 cm. Where should a third charge be placed between them so that it does not experience any force.

Explanation

`Q_1 = 2mu C = 2 xx 10^(-6) C`
`Q_2 = 1 muC = 1 xx 10^(-6)C`
`d = 10 cm`
Let the third charge Q be placed at a distance of X from `Q_1` then x = ?
The resultant force on 3rd charge is zero.
`vec(F_R) = vec(F_1) + vec(F_2) = 0  "    " vec(F_1) + vec(F_2) = 0`
`(Q_1)/(x^2) = (Q_2)/((d - x)^2) implies (2 xx 10^(-6))/(x^2) = (1 xx 10^(-6))/((10 - x)^2)`
By square rooting on both sides
`(1.414)/x = 1/(10 - x) implies x = 5.9 cm " from " 2 mu C`. (or)
Shortcut
`x = d/(sqrt((q_2)/(q_1)) +- 1) = 10/(sqrt(2/1 + 1)) = 10/(2.414)` = 4.1 cm from `1 mu C`
[+ for like charges, - for unlike charges].