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

In summary, we are given a point charge of 6 × 10^-9 C located at the origin and a magnitude of 150 N/C at location ##\langle 0.6,0,0\rangle## m. We are then asked to determine the approximate change in the magnitude of the electric field at the same location when 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. Using the equations ##\vec E=\frac {kq} {r^2}## and ##\vec E_{wire}=\frac {2kqL} {r^3}##
  • #1
Zack K
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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.
 
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  • #2
Zack K said:
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.
Zack K said:
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?
 
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  • #3
haruspex said:
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.

haruspex said:
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.
 
  • #4
Zack K said:
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?
 
  • #5
haruspex said:
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:
 

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  • #6
Zack K said:
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.
 
  • #7
haruspex said:
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.
 
  • #8
Zack K said:
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.
 
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  • #9
haruspex said:
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.
 
  • #10
Zack K said:
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 Ewire 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.
 
  • #11
haruspex said:
Sorry, I said yes too quickly.
The expression you used for Ewire 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.
 
  • #12
Zack K said:
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?
 
  • #13
haruspex said:
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.
 
  • #14
Zack K said:
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.
 
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  • #15
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.)
 
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Related to Change in the Electric Field Due to a Copper Wire Being Inserted

1. How does inserting a copper wire change the electric field?

When a copper wire is inserted into an electric field, it creates a change in the electric field. This is because the copper wire has its own electric field, and when it is inserted into the existing field, the two fields interact and cause a change.

2. What factors influence the change in the electric field?

The change in the electric field due to a copper wire being inserted is influenced by several factors, such as the length and thickness of the wire, the strength of the existing electric field, and the material of the wire. These factors determine the magnitude and direction of the change in the electric field.

3. How does the direction of the electric field change when a copper wire is inserted?

The direction of the electric field changes when a copper wire is inserted based on the direction of the existing electric field and the orientation of the wire. If the wire is inserted parallel to the electric field, the direction of the field remains unchanged. However, if the wire is inserted at an angle, the direction of the electric field will be altered accordingly.

4. What is the significance of the change in the electric field?

The change in the electric field due to a copper wire being inserted is significant because it can affect the behavior of charged particles in the field. This change can also be used to control and manipulate the movement of these particles, making it an important concept in fields such as electronics and electromagnetism.

5. How can the change in the electric field be measured?

The change in the electric field can be measured using an instrument called an electric field sensor. This device can detect and measure the strength and direction of the electric field at a specific point. By comparing the readings before and after the insertion of a copper wire, the change in the electric field can be determined.

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