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Homework Help: Sources of Magnetic Field

  1. Oct 16, 2006 #1
    the question is that:

    A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is [itex] \vec J[/itex]. The current density, although symmetrical about the cylinder axis, is not constant but varies according to the relation

    (the relation is in the attachment)

    where a is the radius of the cylinder, r is the radial distance from the cylider axis, and [itex]I_0[/itex] is a constant haveing units of amperes.
    a) show that [itex]I_0[/itex] is the total current passing through the entire cross section of the wire.
    b). Using Ampere's law, derive an expression for the magnitude of the magnetic field [itex]\vec B[/itex] in the region r>=a .
    c). Obtain an expression for the current I contained in a circular cross section of radius r<=a and centered at the cylinder axis.
    d). Using Ampere's law, derive an expression for the magnitude of the magnetic field [itex]\vec B[/itex] in the region r<=a.


    For a, Since for the entire cross section of the wire, i subt. r=a into the relation. But it will give zero. I shown nothing. If I subt. J=I/A,
    then [itex]I=2 I_0 [1- (\frac{r}{a})^2][/itex]. Anything wrong,
    and how to proof that?
     

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    Last edited: Oct 16, 2006
  2. jcsd
  3. Oct 16, 2006 #2

    siddharth

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    Since the current density is not constant, you need to integrate over the cross section.

    Also, have you taken a look at the https://www.physicsforums.com/showthread.php?t=8997"l? If you post the question that way, you won't need to wait till the attachment is approved.
     
    Last edited by a moderator: Apr 22, 2017
  4. Oct 16, 2006 #3
    i am sorry since i do not familiar that tutorial yet........

    Should i integrate [itex]\frac{2 I_0}{\pi a^2} [1- (\frac{dr}{a})^2][/itex]
    from 0 to a? if yes, how to integerate [itex](dr)^2[/itex]
     
    Last edited: Oct 16, 2006
  5. Oct 16, 2006 #4

    siddharth

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    No, that's completely wrong.

    If you take a small elemental area da, then the current which flows through that bit is [tex]\vec{J}.\vec{da}[/tex]

    To find the net current through the whole wire, in a sense you add up the current through all the small elemental areas.
    So your net current will be

    [tex]I=\int \vec{J}.\vec{da} [/tex]

    Now,
    (i) Can you tell me what elemental area you will take?
    (ii) What will the limits of integration be?
     
    Last edited: Oct 16, 2006
  6. Oct 17, 2006 #5
    elemental area is the small cross section area [itex]dA=2 \pi r da[/itex],
    and the limits of integration is from 0 to a?
     
    Last edited: Oct 17, 2006
  7. Oct 17, 2006 #6
    I have got the ans.
    and the following problems are also be solved,
    thank you so much
     
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