Magenetic field due to wire and cyllinder

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Homework Help Overview

The discussion revolves around calculating the magnetic field due to a wire and a cylinder, focusing on the interactions at a specific point P where the net magnetic field strength is zero. Participants explore concepts from electromagnetism, particularly Ampere's law and the Biot-Savart law, in the context of magnetic fields generated by current-carrying conductors.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss equating magnetic fields from the wire and cylinder at point P, using superposition to find the current in the wire. There are attempts to apply Ampere's law to determine the magnetic field of the cylinder, and questions arise about the interpretation of signs in current direction and the implications of symmetry in magnetic field calculations.

Discussion Status

The discussion is active, with participants providing hints and guidance on using Ampere's law and the principle of superposition. There is an ongoing exploration of the implications of current direction and magnetic field contributions, with some participants questioning their assumptions and interpretations.

Contextual Notes

Participants are navigating through the complexities of magnetic field calculations, including the need to consider the total current within amperian loops and the effects of symmetry on magnetic fields. There is an emphasis on clear communication regarding current direction, avoiding ambiguous signs.

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Homework Statement



Please find it attached.

Homework Equations



0509db4db875447125d72536b672b437.png


The Attempt at a Solution



Ok so at point P the net magnetic field strength is zero, thus if I were to equate the magnetic fields at point P due to the cylinder, and then due to the wire, and sum them together using the principle of superposition and set that result equal to zero solving that for the current on the wire yields the answer tothe first part. Now from the biot savart law I get the magnetic field due to the current on the wire as being :
magcur3.gif

But I'm not sure on how to do the same for the cylinder, any hints?
 

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samjohnny said:
But I'm not sure on how to do the same for the cylinder, any hints?

Have you studied Ampere's law?
 
TSny said:
Have you studied Ampere's law?

Yes we touched on it.
 
Using Ampere's laws will be the simplest way to get the B field of the cylinder.
 
TSny said:
Using Ampere's laws will be the simplest way to get the B field of the cylinder.

Ok, not sure if I'm going about this right. I considered a circular amperian loop centred at the centre of the cylinder with a radius of 2R (so that point P lies at its edge). Then used Ampere's law to ultimately get B=μI/4πR, where I is the total enclosed current, i.e. the current evenly distributed across the cylinder. Is that right?
 
Yes. Good.
 
TSny said:
Yes. Good.

Ah OK excellent, thanks a lot. So after adding them together and equating to zero, I end up with the current in the second wire as being I=-Ic/2 where Ic is the current through the cylinder. Is that Ok?

As for the next bit, I'm not too sure how to proceed with that. The magnetic field at the centre of the cylinder due to the cylinder's own current is given by the biot savart law as B=μIc/2R. Would I then, by the principle of superposition, add on to that the magnetic field at the cylinder's centre due to the wire by considering a circular amperian loop centred about the wire and with a radius of 3R?
 
samjohnny said:
Ah OK excellent, thanks a lot. So after adding them together and equating to zero, I end up with the current in the second wire as being I=-Ic/2 where Ic is the current through the cylinder. Is that Ok?

OK, except what is the interpretation of the negative sign? Is the current in the second wire in the same direction as the current in the cylinder or the opposite direction? (Hint: right hand rule.)

As for the next bit, I'm not too sure how to proceed with that. The magnetic field at the centre of the cylinder due to the cylinder's own current is given by the biot savart law as B=μIc/2R.

This is not the correct value of B at the center of the cylinder due to the cylinder's own current. You should be able to get the correct answer by just using symmetry arguments.

Would I then, by the principle of superposition, add on to that the magnetic field at the cylinder's centre due to the wire by considering a circular amperian loop centred about the wire and with a radius of 3R?

Yes. Be sure to include the direction of the net field.
 
TSny said:
OK, except what is the interpretation of the negative sign? Is the current in the second wire in the same direction as the current in the cylinder or the opposite direction? (Hint: right hand rule.)

I believe it would indicate that the current of the wire flows out of the page, i.e. antiparallel to the direction of the current in the loop.

TSny said:
This is not the correct value of B at the center of the cylinder due to the cylinder's own current. You should be able to get the correct answer by just using symmetry arguments.

Ah I see, due to the symmetry of the circle at its centre, all magnetic field contributions due to the loop itself would cancel each other out to yield a net field of zero at its centre.

TSny said:
Yes. Be sure to include the direction of the net field.

Ok I'll give it a bash.
 
  • #10
samjohnny said:
I believe it would indicate that the current of the wire flows out of the page, i.e. antiparallel to the direction of the current in the loop.

Keep thinking about this.
 
  • #11
TSny said:
Keep thinking about this.

Hmm, I can't seem to figure out how it'd be incorrect. Current is a vector quantity, so surely the negative sign would indicate it flows in the direction opposite to Ic, wouldn't it?
 
  • #12
What is the direction of the magnetic field at point P due to the current in the cylinder?
 
  • #13
TSny said:
What is the direction of the magnetic field at point P due to the current in the cylinder?

Vertically upwards I believe.
 
  • #14
OK. So, what should be the direction of the current in the second wire to give a downward field at P?
 
  • #15
TSny said:
OK. So, what should be the direction of the current in the second wire to give a downward field at P?

Hmm, in the same direction as the current in the loop, into the page. So then should the negative sign be ignored?
 
  • #16
OK. Good. It is best not to use a positive or negative sign to indicate a current direction as it is not necessarily clear to the person trying to interpret your answer. It is much better to use a description such as "into the page".
 
  • #17
TSny said:
OK. Good. It is best not to use a positive or negative sign to indicate a current direction as it is not necessarily clear to the person trying to interpret your answer. It is much better to use a description such as "into the page".

I see now, thank you. For the next part, since the circular amperian loop about the centre of the wire also encompasses a portion of the cylinder, would it be necessary to add the cylinder's current of Ic to the wire's current to obtain the total current within the amperian loop and hence be able to solve for the magnetic field?
 
  • #18
Just use superposition. The net field at the center of the cylinder is the sum of the fields due to the cylinder alone and the second wire alone.
 
  • #19
TSny said:
Just use superposition. The net field at the center of the cylinder is the sum of the fields due to the cylinder alone and the second wire alone.

Ah yes, of course, I've ended up with the magnetic field at the centre of the cylinder as being B=μIc/12πR which is due solely to the contribution of the wire.
 
  • #20
samjohnny said:
Ah yes, of course, I've ended up with the magnetic field at the centre of the cylinder as being B=μIc/12πR which is due solely to the contribution of the wire.

OK. Good work.
 
  • #21
TSny said:
OK. Good work.

Thanks a lot for all your help!
 

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