Where to Place a Negative Charge for Zero Electric Field at the Origin?

[Bearbull24.5]

Homework Statement

A 40 uC charge is placed on x-axis at x=4cm. Where should a negative 60 uC charge be placed to produce a net electric field of zero at the origin?

Homework Equations

Fe=Ke((q1*q2)/r^2)

The Attempt at a Solution

I tried rearranging this equation to solve for r

 

[zhermes]

Not quite. The idea is that there are two charges, the 40 uC charge is at a particular location, you need to figure out where another one must be placed to make the net field zero at the origin.

How do you find the electric field resulting from two charges?

Conceptually, around where would you guess the other charge should be placed?

 

[Bearbull24.5]

Well I know it has to be located further to the right.

To find the Electric field from 2 charges... it wouldn't be as simple as using the equation E=Ke(q/r^2)?

 

[zhermes]

Well I know it has to be located further to the right.

Totally.

To find the Electric field from 2 charges... it wouldn't be as simple as using the equation E=Ke(q/r^2)?

Not quite. That's the field due to a single point charge. Electric fields obey the principle of 'superposition,' however. To find the net field, you simply add the fields from each individual charge.
Formally:
$$E_{net} = \sum_{i=1}^{n} E_i$$
for n charges

 

[rwisz]

, but I am not sure how to take into account the fact that the electric field at the origin should be zero. I think I need to use the principle of superposition to find the distance from the origin where the negative 60 uC charge should be placed. Since the electric field at the origin is the sum of the fields produced by both charges, I can set up the equation:

0 = Ke(40*10^-6*-60*10^-6)/r^2 + Ke(60*10^-6*-60*10^-6)/(r-x)^2

Simplifying this equation, I get:

0 = (2400*10^-12)/r^2 + (3600*10^-12)/(r-x)^2

I am not sure how to solve this equation for r, but I can make some observations. Since both terms on the right side of the equation are positive, the only way for the sum to equal zero is if one term is positive and the other is negative. This means that r and (r-x) must have opposite signs. Also, the larger the value of r, the smaller the value of (r-x), and vice versa. This means that the negative 60 uC charge should be placed closer to the origin than the positive 40 uC charge.

To solve for r, I can set the two terms on the right side of the equation equal to each other and solve for r. This will give me the distance from the origin where the negative 60 uC charge should be placed. However, I am not sure if this is the most efficient or accurate method to use. I would appreciate any suggestions or feedback on my approach.