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Magnetic Field Equation in Spherical Coordinates to Cartesian Coordinates

  1. Jan 31, 2012 #1
    1. The problem statement, all variables and given/known data

    The magnetic field around a long, straight wire carrying a steady current I is given in spherical coordinates by the expression

    [itex] \vec{B} = \frac{\mu_{o} I }{2∏ R} \hat{\phi}[/itex] ,

    where [itex] \mu_{o} [/itex] is a constant and R is the perpendicular distance from the wire to the observation points. Find the expression for [itex]\vec{B}[/itex] in cartesian coordinates.


    2. Relevant equations



    3. The attempt at a solution

    I know I need to get the partial derivative of this with respect to some variable.. but I don't know what that variable is. can someone help me please?
     
  2. jcsd
  3. Jan 31, 2012 #2
    Jacobians are pretty helpful moving between coordinates. You don't necessarily need to go that in depth though. The R is easy. How will the unit vector transform?
     
  4. Jan 31, 2012 #3
    I didn't really get your point.. i mean, is that formula a unit vector?? I think not.

    In terms of R. do u mean I should make it

    R = [itex]\frac{\mu_{o} I}{2∏ \vec{B}}[/itex] [itex] \hat{\phi} [/itex]
     
  5. Feb 1, 2012 #4
    He means the unit vector in your basis.
     
  6. Feb 1, 2012 #5
    basis meaning the observation points..

    its

    R=Rx i + Ry j + Rz k
     
  7. Feb 1, 2012 #6

    vela

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    He means ##\hat{\phi}##.
     
  8. Feb 1, 2012 #7
    hmmm.. now I'm really confused. Can you please tell me the steps on how to do this, then i'll try. I'll show you what I did then you can tell me if I'm wrong or right. Thanks.
     
  9. Feb 1, 2012 #8
    Can someone please help me with this one?? This is the last problem I wasn't able to solve in our problem set. Help will be much appreciated. Thanks.
     
  10. Feb 1, 2012 #9
    Spherical coordinates to cartesian coordinates

    1. The problem statement, all variables and given/known data
    [itex] \vec{B} = \frac{\mu_{o} I }{2∏ R} \hat{\phi}[/itex] , is the equation of Magnetic Field in spherical coordinates. where [itex] \mu_{o} [/itex] is a constant and R is the perpendicular distance from the wire to the observation points. Find the expression for [itex]\vec{B}[/itex] in cartesian coordinates.


    2. Relevant equations



    3. The attempt at a solution

    I tried to equate this equation in terms of [itex]\hat{\phi}[/itex] but after that I'm stuck.. I know also that R being the perpendicular distance from the wire to the observation points will also do the trick but I don't know how it will help. Can someone please help me figure this thing out? help will be much appreciated
     
    Last edited: Feb 1, 2012
  11. Feb 1, 2012 #10

    tiny-tim

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    hi jhosamelly! :smile:

    hint: what shape are the field-lines? :wink:
     
  12. Feb 1, 2012 #11
    Re: Spherical coordinates to cartesian coordinates


    circular.. do you mean I should use equations for circle?
     
  13. Feb 1, 2012 #12

    tiny-tim

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    seems a good idea! :wink:

    what do you get? :smile:
     
  14. Feb 1, 2012 #13
    Re: Spherical coordinates to cartesian coordinates

    [itex] (x-a)^2 + (y-b)^2 = r^2 [/itex]

    or if the center is at the origin


    [itex] x^2 + y^2 = r^2 [/itex]


    how does this help? should I equate this with the R in the equation?
     
  15. Feb 1, 2012 #14

    tiny-tim

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    the centre isn't at the origin, is it?

    it's anywhere along the z-axis :wink:

    and it isn't r, it's R …

    ok, now that you know what shape everything is, write the original formula for B in spherical coordinates :smile:
     
  16. Feb 1, 2012 #15
    Re: Spherical coordinates to cartesian coordinates

    do you mean this??

    [itex] \vec{B} = \frac{\mu_{o} I }{2∏ R} \hat{\phi}[/itex]

    I think this is already in spherical coordinates.

    should i change R now to [itex] x^2 + y^2 [/itex]

    [itex] \vec{B} = \frac{\mu_{o} I }{2∏ (x^2 + y^2)} \hat{\phi}[/itex]
     
  17. Feb 1, 2012 #16

    tiny-tim

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    oops! i meant cartesian coordinates! :redface:
    yes, and finally you need to change phi to x and y (or i and j) :smile:
     
  18. Feb 1, 2012 #17
    Re: Spherical coordinates to cartesian coordinates



    [itex] \vec{B} = \frac{\mu_{o} I }{2∏ (x^2 + y^2)} \hat{\phi}[/itex]

    [itex] \hat{\phi} = - sin \phi \hat{i} + cos \phi \hat{j} [/itex]

    [itex] \vec{B} = \frac{\mu_{o} I }{2∏ (x^2 + y^2)} (-sin \phi \hat{i} + cos \phi \hat{j}) [/itex]

    is this it???? Thanks for your help :)))
     
  19. Feb 1, 2012 #18

    tiny-tim

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    (type \pi for π in latex :wink:)
    that's it! :biggrin:

    (except for the missing square-root :wink:)

    (btw, now that you've got the idea, you don't need to find the shape of the field-lines …

    that was just to help you visualise everything)
     
  20. Feb 1, 2012 #19
    Re: Spherical coordinates to cartesian coordinates

    ow yah.. because its R^2.. hehe.. thanks for the reminder..

    [itex] \vec{B} = \frac{\mu_{o} I }{2\pi (\sqrt{(x^2 + y^2)})} (-sin \phi \hat{i} + cos \phi \hat{j}) [/itex]

    is this ok now or should i also change sin to y/r and cos to x/r ??
     
  21. Feb 1, 2012 #20

    tiny-tim

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    yes! :smile:
     
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