Change in the Electric Field Due to a Copper Wire Being Inserted

[Zack K]

Homework Statement

A point charge of 6 × 10−9 C is located at the origin.
The magnitude magnitude at $\langle 0.6,0,0\rangle$ m is 150 N/C

Next, a short, straight, thin copper wire 5 mm long is placed along the x axis with its center at location $\langle 0.3,0,0 \rangle$ m. What is the approximate change in the magnitude of the electric field at location $\langle 0.6,0,0\rangle$? (Approximate the polarized charges on the short copper wire as a dipole, and when calculating your answer, take the center of the wire as the point at which the net electric field is zero.)

Homework Equations

$\vec E=\frac {kq} {r^2}$
$\vec E_{wire}=\frac {2kqL} {r^3}$

The Attempt at a Solution

I calculated the electric field due to the wire using the 2nd equation. Using q as the charge of the point charge and r as 0.3. Then finding the magnitude of the electric field to the wire should give me the change in the electric field due the wire.

 

[haruspex]

I calculated the electric field due to the wire using the 2nd equation. Using q as the charge of the point charge

What exactly is that equation, i.e. what do the variables in it represent?
It looks like the equation for the field at distance r off to one side from a dipole length 2L with charges q, -q. You do not know what the dipole charges are in the wire, and we are not concerned with a sideways displacement from it. Everything is along the x axis.

The magnitude magnitude at $\langle 0.6,0,0\rangle$ m is 150 N/C

I guess you mean that's the magnitude of the field there due to the point charge, but isn't that deducible from the given charge? Or am I missing something?

 

[Zack K]

What exactly is that equation, i.e. what do the variables in it represent?
It looks like the equation for the field at distance r off to one side from a dipole length 2L with charges q, -q. You do not know what the dipole charges are in the wire, and we are not concerned with a sideways displacement from it. Everything is along the x axis.

Sorry. So k is the constant (9x10^9), q is the charge of charge of an object, in our case q should be the charge of the point charge. L is the length of the wire, and r is a radius between an observation point and the source of a charge. The question doesn't give us any dipole charges.

I guess you mean that's the magnitude of the field there due to the point charge, but isn't that deducible from the given charge? Or am I missing something?

I forgot to mention that when I calculated 150N/C originally, their was no copper wire. The second part of the question introduced the copper wire.

 

[haruspex]

So k is the constant (9x10^9), q is the charge of charge of an object, in our case q should be the charge of the point charge. L is the length of the wire, and r is a radius between an observation point and the source of a charge.

Then I am puzzled as to where this equation comes from. Do you have a reference for it?
It does not seem possible that it is true for r being in any direction from the point charge.

Instead of trusting that, why not just calculate the net field at the centre of the wire in terms of the unknown dipole charges and equate that to zero?

 

[Zack K]

Then I am puzzled as to where this equation comes from. Do you have a reference for it?
It does not seem possible that it is true for r being in any direction from the point charge.

Instead of trusting that, why not just calculate the net field at the centre of the wire in terms of the unknown dipole charges and equate that to zero?

I found it in my textbook "Matters and Interactions" 4th edition. I'll upload the section of the textbook:

 

[haruspex]

I found it in my textbook "Matters and Interactions" 4th edition. I'll upload the section of the textbook:

Ok.
First, the equation is specifically for the case where r is the displacement from the centre of the wire along the line of the wire. This is not as general as you implied in post #3, but does apply here.
Secondly, q in this equation is the dipole charge, not the point charge. They will be different. So you need to find the dipole charge. You can use the method I outlined at the end of post #4.

 

[Zack K]

Ok.
First, the equation is specifically for the case where r is the displacement from the centre of the wire along the line of the wire. This is not as general as you implied in post #3, but does apply here.
Secondly, q in this equation is the dipole charge, not the point charge. They will be different. So you need to find the dipole charge. You can use the method I outlined at the end of post #4.

Right, that makes sense.
So knowing that the net electric field at the center is 0. Would it be right to write it as $\vec E_{charge}=\vec E_{wire} \Longrightarrow \frac {kq_1} {r_1^2}=\frac {2kq_2L} {r_2^3}$. Where $r_1$ and $q_1$ are the distance from the charge to the center of the wire and the charge of the point charge respectively. $r_2$ and $q_2$ is half the length of the wire and $q_2$ is the charge we need to find.

 

[haruspex]

Right, that makes sense.
So knowing that the net electric field at the center is 0. Would it be right to write it as $\vec E_{charge}=\vec E_{wire} \Longrightarrow \frac {kq_1} {r_1^2}=\frac {2kq_2L} {r_2^3}$. Where $r_1$ and $q_1$ are the distance from the charge to the center of the wire and the charge of the point charge respectively. $r_2$ and $q_2$ is half the length of the wire and $q_2$ is the charge we need to find.

Yes.
Edit: no. See my next post.

 

[Zack K]

Yes.

Alright so I calculated for $q_2$ and then plugged it in for the $\vec E_{wire}$ equation and got 600N/C which would be a bad coincidence if I got that whole number so I'm positive it's right. I have some confusion though on what they mean by wanting the change in magnitude of the electric field at 0.6m. That 600N/C is the strength on each side of the wire. Sorry, I'm just hesitant since i have 2 tries left on my online assignment.

 

[haruspex]

Alright so I calculated for $q_2$ and then plugged it in for the $\vec E_{wire}$ equation and got 600N/C which would be a bad coincidence if I got that whole number so I'm positive it's right. I have some confusion though on what they mean by wanting the change in magnitude of the electric field at 0.6m. That 600N/C is the strength on each side of the wire. Sorry, I'm just hesitant since i have 2 tries left on my online assignment.

Sorry, I said yes too quickly.
The expression you used for E_wire in post #7 is for the field the wire of length L creates at a point r₂>>L away in the line of the wire. For the purposes of the equation there you want the field it produces at its own centre. The expression does not apply because r₂ wouid be zero!
There is a very simple way to get the field at the middle of a dipole.

 

[Zack K]

Sorry, I said yes too quickly.
The expression you used for E_wire in post #7 is for the field the wire of length L creates at a point r₂>>L away in the line of the wire. For the purposes of the equation there you want the field it produces at its own centre. The expression does not apply because r₂ wouid be zero!
There is a very simple way to get the field at the middle of a dipole.

Right, so we can use $\vec E=\frac {2kq} {r^2}$ since the electric fields on both ends of the dipole point the same direction if I understood it right.

 

[haruspex]

Right, so we can use $\vec E=\frac {2kq} {r^2}$ since the electric fields on both ends of the dipole point the same direction if I understood it right.

Yes. What relationship does that give between the point charge and the dipole charges?

 

[Zack K]

Yes. What relationship does that give between the point charge and the dipole charges?

That the strength of the dipole charge is twice the strength of the point charge.

 

[haruspex]

That the strength of the dipole charge is twice the strength of the point charge.

Doesn't seem right to me. Write out the equation saying that the net field is zero, being careful with what all the variables mean.

 

[rude man]

First, determine the charge distribution inside the wire. Remember that a Cu wire has zero NET E field. This gives you the dipole.
Then, compute the net effect of the dipole on the field at (.6,0,0).
Since you're only interested in the CHANGE of the E field, does it matter what the field was before the introduction of the wire? (Assume the wire length is << 0.6m.)