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Finding the magnetic field of a loop at far distances

  1. Nov 10, 2016 #1
    1. The problem statement, all variables and given/known data
    Loop of current ##I## sitting in the xy plane. Current goes in counter clockwise direction as seen from positive z axis. Find:

    a) the magnetic dipole moment
    b) the approximate magnetic field at points far from the origin
    c) show that, for points on the z axis, your answer is consistent with the exact field (Ex. 5.6), when z R.

    2. Relevant equations
    ##\vec{m} = I \int d\vec{A}##
    ##\vec{A}_{dip} (\vec{r}) = \frac{\mu_0 \vec{m} \times \hat r}{4 \pi r^2}##
    ##\vec{B}=\nabla \times \vec{A}##

    3. The attempt at a solution
    a) I got that this is ##I\pi R^2 \hat z##
    b) I got ##\vec{A}_{dip} (\vec{r}) = \frac{\mu_0 I R^2}{4 r^2}(\hat z \times \hat r)##, but I don't know how to interpret ##(\hat z \times \hat r)##. I tried taking the vector product by treating them as cylindrical coordinates and using the conversion to cartesian, which resulted in ##\hat \phi##. So it says that the vector potential curls around the z axis, which doesn't make sense...

    Then I know I have to do ##\vec{B}=\nabla \times \vec{A}##, but that's contingent on the previous part being correct.

    c) ???

    Any tips?
     
    Last edited: Nov 10, 2016
  2. jcsd
  3. Nov 10, 2016 #2

    Charles Link

    User Avatar
    Homework Helper

    Hello again. Suggest you google "magnetic dipole". The Wikipedia has a good explanation. One item here that might create some confusion: In some SI units, I have seen the authors write ## B=\mu_o H +\mu_o M ## instead of ## B=\mu_o H +M ##. In any case, try reading Wikipedia. I think the answer is often computed in spherical coordinates rather than cylindrical, but it appears Wikipedia's answers are independent of coordinate system. ## \\ ## editing... Also, there is a formula for the curl of a cross-product of two vectors. It can often be found on the cover or appendix of an E&M textbook. ## \\ ## editing... additional item... A magnetic dipole can also be modeled as a "+" pole at one end and a "-" pole at the other and the inverse square law can be used for the ## H ## from each pole. The calculations using the pole model are a little simpler than Biot-Savart type integrals around the loop. I believe about a month or two ago, I verified the Wikipedia formulas using the pole method when someone else posted a question about the magnetic field of a magnetic dipole. ## \\ ## Here is a "link" to that discussion: https://www.physicsforums.com/threads/equation-of-magnetic-field-produced-by-a-solenoid.888895/
     
    Last edited: Nov 10, 2016
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