Answer
$x = 0.683m$. We only take the positive root.
Work Step by Step
If the net electrostatic force on particle 3 due to particles 1 and 2 is zero,
$\overrightarrow{F_{3 \space net}} = 0$, the x coordinate is unknown
Here we have
$r_{31} = x $
$r_{32} = x – 0.20 m$
Now we equate $ |\overrightarrow{F_{31}}| = |\overrightarrow{F_{32}}| $ and solve for x
$ |\overrightarrow{F_{31}}| = |\overrightarrow{F_{32}}| $
$ \frac{kq_3|q_1|}{(r_{31})^2} = \frac{kq_3q_2}{(r_{32})^2} $
$ \frac{kq_3|q_1|}{(kq_3)(r_{31})^2} = \frac{q_2}{(r_{32})^2} $
$ \frac{|q_1|}{(r_{31})^2} = \frac{q_2}{(r_{32})^2} $
$ \frac{80.0 \mu C}{(x)^2} = \frac{40.0 \mu C }{(x – 0.20 m)^2} $
$ \frac{(x – 0.20 m)^2}{(x)^2} = \frac{40.0 \mu C }{80.0 \mu C} $
$ \frac{(x – 0.20 m)^2}{(x)^2} = \frac{1 }{2} $
Once we have reached at this point, we need to do some algebra.
$STEP \space 1$
Multiply both sides by $x^2$. We neglect unit here for simplicity.
$x^2−0.4x+0.04=0.5x^2$
$x^2 -0.5x^2 −0.4x+0.04= 0 $
$0.5x^2 −0.4x+0.04= 0 $
$STEP \space 2$
Use the quadratic formula to find x
$x=\frac{−b±\sqrt {b^2−4ac}}{2a} $
$x=\frac{−(-0.4±\sqrt {(-0.4)^2−4(0.5)(0.04)}}{2(0.5)} $
$x = 0.683m$. We only take the positive root.
