Question

Suppose the charge q2 in the figure can be moved left or right along the line connecting the charges q1 and q3. Given that q = +17

Answer 1

Suppose distance between q1 and q3 is L and that between q1 and q2 is x (you know L and you need to find out x)
Now since q2 can only move along the line between q1 and q3, the distance between q2 and q3 will be (L-x)
Force between q1 and q2: q1*q2/(k*x^2) where k is 1/(4*pi*e)...since q1 = +1 and q2 = -2: the force on q2 is attractive and directed from q2 towards q1.
Similarly, force between q1 and q3 is: q1*q3/(k*(L-x)^2) and is attractive and directed from q2 towards q3.
Thus the two forces are acting opposite to each other and you need to find the condition (for x) when the two are equal:
Thus just equate the 2 forces:
-2/x^2 = -6/(L-x)^2 (by plugging the values of q1, q2 and q3)
Thus: (L-x)^2 = 3x^2
or, L^2 -2Lx + x^2 = 3x^2
or, 2x^2 + 2Lx - L^2 = 0

The solutions are x = 0.366L and -1.366L
L = 32 cm

Thus the distance of q2 from q1 would be: + 11.712cm implying that q2 lies between q1 and q3 (i.e. q1 lies on one side of q2 and q3 lies on the other side)...(arrangement is: q1_11.712cm_q2_20.288cm_q3 written in the style: charge followed by distance separation and then another charge)
The other distance could be: L = -43.712cm implying that q2 lies on the same line as q1 and q3, but both q1 and q3 lie on the same side (i.e. q2 is not in between the charges)...(the arrangement is: q2_43.712cm_q1_32cm_q3 OR q3_32cm_q1_43.712cm)