Electric field on each midpoint of square of side L

[Dexter09]

θ

Homework Statement

What would be the electric field on the midpoint of each side of a square with point charges of +Q on diagonal corners and -Q on diagonal corners?

Homework Equations

E=kQ/r^2 Ex=Ecosθ Ey=Esinθ

The Attempt at a Solution

Considering the midpoint of the side at the bottom of the square, I found the x components of each of the four electric fields it would have (with each of the four corners of the square).
E1x = E1 = -4kQ/L^2
E2x= E2 = 4kQ/L^2
E3x=E3cosθ= -4kQ/5L^2(cosθ)
E4x=E4cosθ=4kQ/5L^2(cosθ)

When I find the sum of the x components, it is zero. I found the y-components in a similar way, and believe they will cancel out, too. So, I think the electric field on the midpoint should be zero, but the solution in the back of the book indicates that it is 1.15x10^10 Nm^2/C^2 (Q/L^2) in the x-direction.
I haven't done the calculation for each midpoint, but I feel like each one would cancel out in the same way, but the solution says that each one is the answer listed above in either the x or y direction.

 

[ap123]

Hello Dexter09

I'm a bit confused about how you worked out the components.
How did you get the values for E1x and E2x?
Edit : I mean the directions

 

[Dexter09]

I had the midpoint with -Q on the left, and +Q on the right. I drew my E1 vector from the midpoint toward +Q, and the E2 vector from the -Q toward the midpoint.

 

[Dexter09]

And since the vectors were completely horizontal, there is no y component. So the x-components are equal to the total E for each.

 

[rude man]

θ

Homework Statement

What would be the electric field on the midpoint of each side of a square with point charges of +Q on diagonal corners and -Q on diagonal corners?

Homework Equations

E=kQ/r^2 Ex=Ecosθ Ey=Esinθ

The Attempt at a Solution

Considering the midpoint of the side at the bottom of the square, I found the x components of each of the four electric fields it would have (with each of the four corners of the square).
E1x = E1 = -4kQ/L^2
E2x= E2 = 4kQ/L^2
E3x=E3cosθ= -4kQ/5L^2(cosθ)
E4x=E4cosθ=4kQ/5L^2(cosθ)

When I find the sum of the x components, it is zero. I found the y-components in a similar way, and believe they will cancel out, too. So, I think the electric field on the midpoint should be zero, but the solution in the back of the book indicates that it is 1.15x10^10 Nm^2/C^2 (Q/L^2) in the x-direction.
I haven't done the calculation for each midpoint, but I feel like each one would cancel out in the same way, but the solution says that each one is the answer listed above in either the x or y direction.

Taking the top two charges for example, each corner charge exerts its force in the same direction on a unit test charge placed at the mid-point.

So the E field = 2E1x or 2E2x depending on which corners you placed the + and - charges.

 

[Dexter09]

Actually, I was a little unsure of how to treat the q for the midpoint since I didn't think it would have a net charge. I just used Coulomb's Law and made E = kQ/r^2 (and since the r is L/2, it became E=4kQ/L^2)

 

[ap123]

Since you are calculating the electric field and not the electric force, then you do not need to consider a charge at the midpoint.

Just use the equation for the electric field produced by a point charge.
If you draw a diagram with the directions of the fields at your point, then you will see that the fields will not cancel.

 

[rude man]

Actually, I was a little unsure of how to treat the q for the midpoint since I didn't think it would have a net charge. I just used Coulomb's Law and made E = kQ/r^2 (and since the r is L/2, it became E=4kQ/L^2)

Using Coulomb's law correctly you get the right answer. Look at the direction of the field at the mid-point due to the two charges q and -q.

It's not a question as to the value of the "net charge" at the mid-point. There is no charge there.

 

[Dexter09]

I had +Q(1) in the upper left and lower right, and -Q(2) - in upper right and lower left. I was considering the midpoint on the bottom of the square (with -Q(3) on its left and +Q(4) on its right. I drew vectors E1 from -Q(3) to the midpoint, vector E2 from the midpoint toward +Q(4), and E3 on a diagonal from +Q(1) to the midpoint, and E4 from the midpoint to -Q(2). Does my diagram make sense?

 

[ap123]

Your directions are not correct for the electric fields.

If I have a positive charge, does the electric field created by the charge point away or towards it?

 

[Dexter09]

Now I am completely confused. Should I even be drawing electric field vectors from the midpoint, if it is not a charge?

 

[Dexter09]

Away from positive, toward negative.

 

[ap123]

hould I even be drawing electric field vectors from the midpoint, if it is not a charge?

You should draw the field vectors from the midpoint because that is the point where you want to evaluate the electric field.

Away from positive, toward negative.

Yes

So, start by drawing the field vectors at the midpoint due to the 2 charges on either side.
You should find that the vectors point in the same direction.

 

[Dexter09]

So E1 and E2 both go from the midpoint toward the -Q corner that is to their left. And E3 and E4 each go from the midpoint on diagonals in the positive x direction (their y components cancel out). I got E1x=-4kQ/L^2, E2x=-4kQ/L2, E3x=E4x=4sqrt5kQ/25L^2
But the sum of all the x-components still does not add up to 1.15 x 1-^10. I keep checking my math, but I can't find where I am going wrong.
maybe I should separate it into two problems? A point at the apex of a triangle formed with other vertices of +Q and -Q, and then a point on a line equidistant between +Q and -Q. Does that sound reasonable?

 

[rude man]

So E1 and E2 both go from the midpoint toward the -Q corner that is to their left. And E3 and E4 each go from the midpoint on diagonals in the positive x direction (their y components cancel out). I got E1x=-4kQ/L^2, E2x=-4kQ/L2, E3x=E4x=4sqrt5kQ/25L^2
But the sum of all the x-components still does not add up to 1.15 x 1-^10. I keep checking my math, but I can't find where I am going wrong.
maybe I should separate it into two problems? A point at the apex of a triangle formed with other vertices of +Q and -Q, and then a point on a line equidistant between +Q and -Q. Does that sound reasonable?

Look again at your E3x and E4x. Pay attention to the direction of the E field due to the two bottom charges on the top middle location.

 

[rude man]

But the sum of all the x-components still does not add up to 1.15 x 1-^10. I keep checking my math, but I can't find where I am going wrong.
maybe I should separate it into two problems? A point at the apex of a triangle formed with other vertices of +Q and -Q, and then a point on a line equidistant between +Q and -Q. Does that sound reasonable?

No. See my previous post.

 

[Dexter09]

Aighh. I took so long to type that I was logged out. Here is a quick recap:
E1x=4kQ/L^2 = 3.60 x 10^10 Q/L^2
E2x = -4kQ/L^2 = -3.60 x 10^10 Q/L^2
E3x = E3 sin theta = 4kQsqrt5Q/25L^2 = 3.22 x 10^9 Q/L^2
E4x = E4 sin theta = -4kQsqrt5Q/25L^2 = -3.22 x 10^9 Q/L^2
so the x components cancel
E1y = 0
E2y = 0
E3y= E3 cos theta = -8sqrt5Q/25 L^2= -6.43 x 10^9 Q/L^2
E4y=E4 cos theta = -8sqrt5Q/25 L^2= -6.43 x 10^9 Q/L^2

so the y component = 1.29 x 10^10Q/L^2

This give a magnitude of 1.29 x 10^10 Q/L^2 to the electric field in the y direction. but I can tell there is something wrong with my signs. From my diagram I can see that both E1 and E2 are in the negative x direction, and the E3x and E4x are in the positive x direction, and E3y and E4y are equal but opposite and should cancel. But I feel like I am using the Coulomb's law formula correctly... ?

 

[ap123]

The values look ok, but the signs are messed up
You had the correct signs in post 14.

My answer doesn't agree with the answer you quoted. Could you check the book answer again.

 

[Dexter09]

The book answer is: 1.15 x 10^10 Q/L^2 for two midpoints, and -1.15 x 10^10 Q/L^2 for the other two midpoints. I've tried my numbers in different combinations to see if I can find my sign error, but I just can't figure it out. Coulomb's Law has a +Q in E1 and a -Q in E2, but they both point in the same direction, so it seems to me that they should each have negative x components. E3 and E4 also have opposite signs based on Coulomb's law, but they both point in the positive x direction, so it seems like they should each have a positive x-component. But when I add all the x components together I get -6.55 x 10^10 Q/L^2.
Thanks for all of your help. I wish I could figure it out - I really like working on these problems. I thought I was understanding, but I guess I must be missing something.

 

[ap123]

The first thing to do is to make sure you have the directions correct for the electric field vectors.
Consider the midpoint of the bottom line, (I've got +Q at top-left and bottom-right and -Q at top-right and bottom-left).
What are the directions of the 4 electric field vectors at that point?
(I've got the positive x-axis pointing right and the positive y-axis pointing up).

Remember : E points away from a positive source charge and towards a negative one

 

[rude man]

You need to update your expressions for E1x, E2x, E3x and E4x for us. Also your latest on where the +Q and -Q charges are located.

 

[Dexter09]

The Qs are located as described in post 20. In relation to the midpoint of the bottom side of the square, I have vector E1 going from the bottom right +Q toward the midpoint, E2 going from the midpoint to the lower left -Q,E3 extending from the midpoint on an angle in the positive x/negative y direction, and E4 extending from the midpoint up to the upper right -Q (E3 and E4 reflect across the x-axis). The expressions are as in post 17. My instinct tells me that both E1x and E2x should be negative, since they both point in the negative x direction...but using Coulomb's Law I don't see how I can make E1x negative since it is extending from +Q.

 

[ap123]

Coulombs law cannot be applied to this situation since there is no charge at the midpoint.

both E1x and E2x should be negative, since they both point in the negative x direction

Yes.

What about the other 2 field vectors?

 

[rude man]

The Qs are located as described in post 20. In relation to the midpoint of the bottom side of the square, I have vector E1 going from the bottom right +Q toward the midpoint, E2 going from the midpoint to the lower left -Q,E3 extending from the midpoint on an angle in the positive x/negative y direction, and E4 extending from the midpoint up to the upper right -Q (E3 and E4 reflect across the x-axis). The expressions are as in post 17. My instinct tells me that both E1x and E2x should be negative, since they both point in the negative x direction...but using Coulomb's Law I don't see how I can make E1x negative since it is extending from +Q.

Ex1 and Ex2 are incorrect in post 17.

You badly need to understand what an electric field is: the force on a unit positive test charge. E = kQ*Q_test/r^2 where Q_test = 1 Coulomb. So put a Q = 1 charge at each midpoint, ONE AT A TIME, and determine the magnitude and direction of the x and y force components due to the four corner charges at that midpoint.

Then move that test charge to each of the other three midpoints, ONE AT A TIME.

Then update us on your computations for the field at the bottom midpoint.

 

[Dexter09]

Ok, now I just feel stupid and defeated. Thanks again for trying to help me.

 

[Dexter09]

I think that both of theE3 and E4 x components are in the positive x direction. Are you able to tell me if I have the vectors at least pointing in the right directions as I described them in post 22? I would try to do what rude man suggested in post 24, but I don't understand how the midpoint interacts with the +Q corners...did he mean that I had the vectors drawn incorrectly?

 

[ap123]

I think that both of the E3 and E4 x components are in the positive x direction

Yes, that's correct.
So, you have the correct sign for all the x-components.
(E1x and E2x negative, E3x and E4x positive)

I agree with the magnitudes you obtained in post #17.
But this doesn't agree with the book answer - maybe someone else can check.

rude_man is suggesting you use the fundamental definition of the electric field, which is the force on a unit positive charge, ( consult your book/notes if you need reminding of this).
If you want to look at this method, then put a charge of +1C at the midpoint and use Coulomb's law to work out the force on the +1C charge. The direction of the force is the same direction as the electric field at that point.

 

[rude man]

I agree with the magnitudes you obtained in post #17.
But this doesn't agree with the book answer - maybe someone else can check.

I get E1x = E2x = -8kQ/L^2 = -7.2e10Q/L^2
E3x = E4x = +1.44e10Q/L^2

 

[ap123]

I get E1x = E2x = -8kQ/L^2 = -7.2e10Q/L^2
E3x = E4x = +1.44e10Q/L^2

I'm probably missing something obvious, but how did you get the 8 in E1x and E2x?

 

[rude man]

I'm probably missing something obvious, but how did you get the 8 in E1x and E2x?

E1x = -kQ/(L/2)^2 = -4kQ/L^2
E2x = same
E1x + E2x = -8kQ/L^2

 

[ap123]

OK - this is the same as I've calculated then.

For E3x + E4x, I'm getting (8√5)/25 for the coefficient.
This multiplied by k gives me 6.43e9 - quite a lot different from your value.

Edit : the difference between the 2 values is √5

 

[rude man]

OK - this is the same as I've calculated then.

? You said you agreed with to the magnitudes of post 17 ...

For E3x + E4x, I'm getting (8√5)/25 for the coefficient.
This multiplied by k gives me 6.43e9 - quite a lot different from your value.

E3x = E4x = +(8/5)kQ/L^2
The entire Ex field at the bottom mid-point is -6.4kQ/L^2 = -5.76e10Q/L^2.

Edit : the difference between the 2 values is √5

Don't understand that ...

 

[ap123]

? You said you agreed with to the magnitudes of post 17 ...

I do.
You wrote that E1x = E2x = -8kQ/L^2 rather than E1x + E2x = -8kQ/L^2 which is what I was unsure about.
This is also the magnitude that is in post #17

E3x = E4x = +(8/5)kQ/L^2

You mean
E3x + E4x = +(8/5)kQ/L^2
I have an additional √5 in the denominator

Don't understand that ...

See above

 

[rude man]

OK. Right on both counts. I meant "+" when I said "=".

Where did you get the extra sqrt(5)? The distance squared from the bottom midpoint to either upper corner is r^2 = L^2 + (L/2)^2 = 5L^2/4 by Pythagoras.

 

[ap123]

Where did you get the extra sqrt(5)? The distance squared from the bottom midpoint to either upper corner is r^2 = L^2 + (L/2)^2 = 5L^2/4 by Pythagoras.

From the cosθ when I took the x-component of the field.
cosθ = 1 / √5

 

[rude man]

From the cosθ when I took the x-component of the field.
cosθ = 1 / √5

If you're going to do it that way, isn't it cos²θ?

Never mind, I just woke up. Yes, the cosθ needs to be incorporated in my Ex3 and Ex4, making it Ex3 + Ex4 = 0.447(8/5)Q/L^2. Pardon my obtuseness.

 

[ap123]

If you're going to do it that way, isn't it cos²θ?

How do you get that?
I'm just taking the x-component of the vector, eg
E3x = E3 * cosθ

Edit :

Pardon my obtuseness

OK

I hope the OP has absorbed this long thread.

 

[rude man]

How do you get that?
I'm just taking the x-component of the vector, eg
E3x = E3 * cosθ

Edit :

OK

I hope the OP has absorbed this long thread.

So at least you & I agree, right?

 

[ap123]

So at least you & I agree, right?

Yes :)