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## Homework Statement

Two point charges ##Q_1 = 9 \mu##C and ##Q_2 = -16 \mu##C are fixed in space on a distance r=7cm. At what distane ##x_1## from the first charge, and ##x_2## from the second charge, should we place the third charge ##Q_3## so that net force on ##Q_3## is zero? make a sketch with force diagram for third charge. Is ##Q_3## positive or negative? What distance from ##Q_1'## and ##Q_2## should we put ##Q_3'## at, if charge ##Q_3'## is ##0.5Q_3##

## Homework Equations

##Q_1 = 9 \mu C=9 \cdot 10^{-6}##C

##Q_2 = -16 \mu C=-16 \cdot 10^{-6}##C

r=7cm

##Q_3 =?##

## The Attempt at a Solution

Charge

**##Q_3## must be positive**in order to be stationary in presence of charges ##Q_1## and ##Q_3##. It's visible from the sketch in the attachment.

Since the net charge on ##Q_3## is zero it follows that ##F_{13}=F_{23}##.

##F_{13}=k \cdot \frac{Q_1 Q_3}{x_{1}^2}##

##F_{23}=k \cdot \frac{Q_2 Q_3}{x_{2}^2}##

##x_2=r+x_1## is visible from the picture attached.

Now I substitute ##F_{13}## and ##F_{23}## in the first equation with corresponding equations:

##k \cdot \frac{Q_1 Q_3}{x_{1}^2}=k \cdot \frac{Q_2 Q_3}{x_{2}^2}##

I divided it by ##k \cdot Q_3##:

##\frac{Q_1}{x_{1}^2}=\frac{Q_2}{x_{2}^2}##

It is visible from this equation that no matter what is the value for ##Q_3## the distance from ##Q_1## and ##Q_2## won't be affected since there is no ##Q_3## in this formula. I can conclude that

**charge ##Q_3'## put in place of ##Q_3## will be at the same distance from charges ##Q_1## and ##Q_2##.**

I substitute ##x_2## with ##r+x_1##:

##\frac{Q_1}{x_{1}^2}=\frac{Q_2}{(r+x_1)^2}##

##Q_1 \cdot (r+x_1)^2=Q_2 \cdot x_{1}^2##

##Q_1 \cdot (r+x_1)^2-Q_2 \cdot x_{1}^2=0##

##9 \cdot 10^{-6} \cdot (r^2+2r x_1+x_1^2)-(-16) \cdot 10^{-6} \cdot x_{1}^2=0##

##9 \cdot 10^{-6} \cdot (r^2+2r x_1+x_1^2)+16 \cdot 10^{-6} \cdot x_{1}^2=0~~~~/\cdot 10^{-6}##

##9 \cdot r^2+9 \cdot 2r x_1+9 \cdot x_1^2+16 \cdot x_{1}^2=0##

##25 \cdot x_{1}^2+18 \cdot r x_1+9 \cdot r^2=0##

If I try to solve this equation with r=7cm I get a result with real and complex part. If I insert r=0.07m I get something similar. How do I fix this?