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## Homework Statement

Point charges ##q_1=50 \mu C## and ##q_1=-25 \mu C## are placed 1.0 m apart.

Where must a third charge ##q_3=20 \mu C## be placed so that the net force on it is zero?

## Homework Equations

$$\vec F=\frac{1}{4\pi {\varepsilon}_0}\frac{|q_1q_2|}{r^2}\hat r$$

## The Attempt at a Solution

Let ##r_1## be the distance from ##q_1## to ##q_3##, and ##r_2## be the distance from ##q_2## to ##q_3##

$$\vec F=0=\frac{1}{4\pi {\varepsilon}_0}(\frac{|q_1q_3|}{{r_1}^2}+\frac{|q_3q_2|}{{r_2}^2})

\\ -\frac{|q_2q_3|}{{r_2}^2}=\frac{|q_3q_1|}{{r_1}^2} \Rightarrow -\frac{|q_2|}{{r_2}^2}=\frac{|q_1|}{{r_1}^2}

\\ -{r_1}^2|q_2|={r_2}^2|q_1|$$

Plug in values:

$$-{r_1}^2|q_2|={r_2}^2|q_1| \Rightarrow -{r_1}^2(25 \mu C)={r_2}^2(50 \mu C)

\\ -{r_1}^2=2{r_2}^2$$

I am not sure if I missed something in my approach, but I do not know how to continue.

Thank you in advance