having trouble with the following
There are three point charges along the x-axis:
q1 is +3µC, located at the origin
q2 is -5µC, located at x = +0.200m
q3 is -8µC
Where is q3 located if the net force on it is 7.00 N in the -x direction?
The way I see to go about this is as follows:
It is not located at x>0.2, as q2 repulses it and is closer and larger than q1
At 0 < x < 0.2 the force from both q1 and q2 goes in -x direction
At x < 0 It is repulsed by q2 but attracted by the closer q1. Given q2's larger size it may still be in this range
If q3 is at x < 0 (which I know to be correct), then F1 is +x direction and F2 is -x direction, so the problem is expressed as F1 - F2 = -7 N where F1 is the force from q1 on q3 and F2 is the force from q2 on q3.
By Coloumb's Law we have that:
F1 = k(q1q3)/sq(r1) and F2 = k(q2q3)/sq(r2)
r1 equals -x where x is the location of q3 on the x-axis (negative), and r2 equals -x+0.2
F1 - F2 = -7
k(q1q3)/sq(-x) - k(q1q3)/sq(-x+0.2) = -7
k(q1q3) and k(q2q3) are known numbers, and gives:
(2.16*10e-1)/sq(-x) - (3.60*10e-1)/sq(-x+2) = -7
Now in earlier problems in the book steps were simple from here, as the right side was 0; so you could just multiply the equation with sq(-x+2)*sq(-x) and solve the polynomial for x = 0. In this case though that leaves a fourth power polynomial on the right side since it isn't empty, which I have no idea how to practically solve.
Also, plotting the left-side equation I get here into a graph program and identifying F=-7 gives me an x value slightly different from the book (at -0.161, while the book says the solution is -0.144); larger than any rounding errors I can think of would imply.
Basically I've been scratching my head over this for a few hours and can't seem to figure out what I am misunderstanding, if it's the mathematics or some aspect of the physics of it that I'm not getting.