Finding the Total Electric Field at X=0 on the X-Axis

[eestep]

Homework Statement

A charge of -11.8 micro-Coulombs is placed on x-axis at x=0 meters. Another charge of -7.3 micro-Coulombs is placed on x-axis at x=+.42 meters. Other than infinity, where on x-axis is total electric field equal to zero? Answer in centimeters.
q₁=-11.8$$\mu$$C
q₂=-7.3$$\mu$$C

Homework Equations

E=kq/r²$$\widehat{}r$$

The Attempt at a Solution

E₁=E₁$$\widehat{}x$$=-(8.99*10⁹9Nm²/C²)(11.8*10^-6C)/r₁²$$\widehat{}x$$
E₂=E₂$$\widehat{}x$$=(8.99*10⁹Nm²/C²)(7.3*10^-6C)/r₂²
E_x=E_1x+E_2x=-(8.99*10⁹Nm²/C²)(11.8*10^-6C)/r₁²+(8.99*10⁹Nm²/C²)(7.3*10^-6C)/r₂²=0
11.8/7.3=r₁²/r₂^2
r₁/r₂=$$\sqrt{}11.8/7.3$$

 

[0ddbio]

I didn't check all of your steps. But if that is right then what you want now is to just have the x-coordinate. So really you only want one 'r', the other one can be written in terms of this r as say, r2 = 0.42 - r1.
Then you can just solve for your r which is the x-coordinate of the equilibrium point.

 

[eestep]

How do I do that? I get r2^2=7.3r1^2/11.8.

 

[0ddbio]

Well if the ratio is correct:
$$\frac{r_{1}}{r_{2}} = \frac{\sqrt{11.8}}{7.3}$$
Just look at the problem in question. Your first charge is at x=0 the second charge is at x=0.42
Now you are asked to find the equilibrium point that should just be a single coordinate. So from the origin to the equilibrium point is r let's say, this also happens to be the distance from the first charge as well since the first charge happens to be at the origin which you called $r_{1}$. So you have a relation between the distance from the first charge to the equilibrium point and the coordinate of the equilibrium point... which is just: $r_{1} = r$.
Now you want to express the distance from the second charge to the equilibrium point. It should be obvious that it is: $r_{2} = 0.42 - r$ which is the same as: $r_{2} = 0.42 - r_{1}$

If it's not obvious then draw a picture. Put the equilibrium point between the two charges and come up with those expressions for the distances to each charge, where r is the unknown x coordinate of the equilibrium point.

So therefore your ratio becomes:
$$\frac{r}{0.42 - r} = \frac{\sqrt{11.8}}{7.3}$$
Then you just do some algebra and solve for r

you should get r = <some number> that does not depend on $r_{1}$ or $r_{2}$.

 

[eestep]

I appreciate your aid!