# Electric Field

## 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
Ex=E1x+E2x=-(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=$$\sqrt{}11.8/7.3$$

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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.

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

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.

$$\frac{r}{0.42 - r} = \frac{\sqrt{11.8}}{7.3}$$
you should get r = <some number> that does not depend on $r_{1}$ or $r_{2}$.