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Biot-Savart's Law for cylindrical conductor

  1. Apr 29, 2016 #1
    1. The problem statement, all variables and given/known data
    The figure shows a cross section across a diameter of a long cylindrical conductor of radius a = 2.92 cm carrying uniform current 151 A. What is the magnitude of the current's magnetic field at the center of the conductor?
    HW9Q8.png

    2. Relevant equations
    Biot-Savart's Law

    3. The attempt at a solution
    B for circular loop = ui/2r
    B = ui/2a
    B = 0.0032 T
    Which is the wrong answer, what happened?
     
  2. jcsd
  3. Apr 29, 2016 #2

    TSny

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    This formula is for the magnetic field at the center of a circular loop of current. But in your problem the current is not flowing in a circular loop. It is flowing in a long, straight, cylindrical conductor.
     
  4. Apr 30, 2016 #3
    Is this still do-able with the Biot Savart law?
     
  5. Apr 30, 2016 #4

    TSny

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    Yes, you can easily find B at the central axis by using the Biot-Savart law and symmetry.

    However, if you want to find B for an arbitrary value of r in the picture, then it would be easier to use another law.
     
  6. Apr 30, 2016 #5
    In this case, I think the current is flowing through the center. Then that means there is no field at the center right?
     
  7. Apr 30, 2016 #6

    TSny

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    I'm not understanding this argument. Can you elaborate? The current is flowing at all points of the cylinder, not just along the central axis.
     
  8. Apr 30, 2016 #7
    My bad, then can you show me how to do this problem? We have only learned so far the law for wires.
     
  9. Apr 30, 2016 #8

    TSny

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    The problem only asks for the B field at the center axis of the cylinder. For this, you can just use symmetry arguments. Think of the total current distribution as made up of a lot of long, parallel, straight filaments of current. Each filament is like a thin, straight wire carrying current. Use what you know about the direction of the B field due to a long, straight wire.
     
  10. Apr 30, 2016 #9
    For parallel wires carrying the same current, there will be no net magnetic field at a point between them. So can I generalize this to the situation here?
     
  11. Apr 30, 2016 #10

    TSny

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    Yes. Good.
     
  12. Apr 30, 2016 #11
    Now the problem asks for the field at radial distance 1 cm. How can I use symmetry for this?
     
  13. Apr 30, 2016 #12

    TSny

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    Now you need to do some math! The law of choice would be Ampere's law, not the Biot-Savart law. Symmetry will still be important.
     
  14. Apr 30, 2016 #13
    Edit: I will come back to this
     
    Last edited: Apr 30, 2016
  15. Apr 30, 2016 #14
    Just watched a quickie on Ampere's Law. So I'm getting this:
    ui = ∫B⋅dl
    ui = B∫dl from 0 to 2πr
    ui/2πr = B = 0.0030 T

    Edit: I'm missing a current ratio?
     
  16. Apr 30, 2016 #15
    Current per Area = 151 / (.0292^2 * pi) = 5.637 x 10^4 A/m^2
    Current for Area of 1 cm radius = 5.637 x 10^4 A/m^2 * pi(.01^2)
    = 3.141592653589794e-04

    Use this for ampere's law gets:
    3.5420e-04 T
     
  17. Apr 30, 2016 #16

    TSny

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    Looks good.
     
  18. Apr 30, 2016 #17
    Now at, the wire's surface, It encloses the total current, at the radius of the wire. From the Ampere's Law, I get 0.0010T which is the wrong answer?
     
  19. Apr 30, 2016 #18

    TSny

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    I think that's the right answer. (Unless you need to get the number of significant figures correct also.)
     
  20. Apr 30, 2016 #19
    I was marked wrong, I'll get back after asking the professor
     
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