Electrostatics (four equal point charges)

In summary, to balance all the charges in a square, a charge of -15.839 μC must be placed in the center.
  • #1
Antonius
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Homework Statement


Four equal point charges q = +22.4 μC sit at each corner of a square. What charge Q, in μC, must be placed in the center so all the charges are in equilibrium? Your answer must include the sign of the charge.

Homework Equations



F = q1*q2 * k / r^2

3. The Attempt at a Solution [/B]

My attempt was to choose one corner (let's say q3), try to find the electrostatic force between q3 and other charges (q1,q2,q4), then add all the y and x components of forces acting to q3, then find the magnitude. Then make the magnitude equal to force acting from the center to q3.

I don't need someone solving this problem for me, I just need a hint to start with. Please check my logic and see if there is any flow. Because I solved it this way once and I had an equation with both q and x (distance) in one equation...
 

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  • #2
Anvar said:
My attempt was to choose one corner (let's say q3), try to find the electrostatic force between q3 and other charges (q1,q2,q4), then add all the y and x components of forces acting to q3, then find the magnitude. Then make the magnitude equal to force acting from the center to q3.
Sounds like a good plan to me.
 
  • #3
But I am ending up with two unknowns in one equation...
 
  • #4
Antonius said:
But I am ending up with two unknowns in one equation...
What are your unknowns?
 
  • #5
Doc Al said:
What are your unknowns?

I am taking the side of the square as "r" [so half is r/2] and 4 of the charges are all the same, q, and last (fifth, in the center) charge is Q.

After doing what I've said in the original post, my equation (after all the simplification) is 4 * r^2 * q + √(2) * q = 8 * r^2 * Q

:/ It's given that q is + 22. 4 C, but, r and Q are not given... my original task is to find Q so that everything will be balanced (equilibrium.)
 
  • #6
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Last edited:
  • #7
Antonius said:
After doing what I've said in the original post, my equation (after all the simplification) is 4 * r^2 * q + √(2) * q = 8 * r^2 * Q
Rethink this equation. All terms must have the same units for an equation to make sense. (When you get it right, the r will cancel.)
 
  • #8
I can't get rid off that one "r^2" ...

Checked my algebra thrice. Should be alright. Do you really think the logic was fine? Where else the problem might be... :/
 
  • #9
Antonius said:
I can't get rid off that one "r^2" ...

Checked my algebra thrice. Should be alright. Do you really think the logic was fine? Where else the problem might be... :/
Show each step. List the force you get from each charge.
 
  • #10
Doc Al said:
Show each step. List the force you get from each charge.
Before I do that. Can you please check these equations (whether they are set up correctly or no):

( I attached the photo of the image)

F 1 due 2 = (k q^2 / r^2) i
F 1 due 3 = (k q^2 / 4 r^4) i

F 1 due 4 = (k q^2 / r^2) j
F 1 due 3 = (k q^2 / 4 r^4) j

F 1 due 5 = 4 k q Q / r^4

My logic: Add all the i's and add all the j's. Square both i and j and take the square root and equal the answer to F 1 due 5.

:/
 

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  • #11
Antonius said:
F 1 due 3 = (k q^2 / 4 r^4) i
How did you get an r^4? That's where you are going wrong.
 
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  • #12
Doc Al said:
How did you get an r^4? That's where you are going wrong.

Well, q3 lies on the opposite (diagonally) direction from q1. So I needed "r" [Electrostatic force between two charged particles: F = k q1 q 2 / r^2] So to find it I used pythagorean theorem [ two sides are equal since it's a square ] so the diagonal ( the distance between q1 and q3) = r^2 + r^2 = 2 r^2 BUT when I plug it in the formula I need to raise it to another square so it's (2r^2)^2 = 4 r^4 ... Am I wrong?
 
  • #13
What am I talking about. D:

Forget it.

So silly.

BRB
 
  • #14
Antonius said:
so the diagonal ( the distance between q1 and q3) = r^2 + r^2 = 2 r^2
The diagonal squared is 2 r^2.

Antonius said:
BUT when I plug it in the formula I need to raise it to another square so it's (2r^2)^2 = 4 r^4 ... Am I wrong?
Yes, you are wrong. :smile:
 
  • #15
Aha! Such a careless mistake!

I got Q = √(q^2/2 = - 15.839 μC

And the negative answer is the one I want. :)

Hopefully it's correct.

Thank you very much, Doc Al!
 

1. What is electrostatics?

Electrostatics is the study of stationary electric charges and the forces and fields they create.

2. What is a point charge?

A point charge is an idealized model of an electric charge that has no physical size or structure, and is considered to be located at a single point in space.

3. What happens when four equal point charges are arranged at the corners of a square?

When four equal point charges are arranged at the corners of a square, they create an electric field that is directed towards the center of the square. The electric field at the center is zero, meaning there is no net force acting on a small test charge placed at that point.

4. What is the net electric field at the center of the square?

The net electric field at the center of the square is zero, as the electric fields created by each of the four point charges cancel each other out.

5. How does the distance between the point charges affect the electric field at the center of the square?

The electric field at the center of the square is directly proportional to the distance between the point charges. As the distance increases, the electric field decreases, and vice versa.

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