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Dipole of Magnetic field in polar coordinates

  1. Aug 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi everybody... i have a bad problem with my brain:

    starting from the Vectorial form of the magnetic dipole:

    [itex] \vec{B}(\vec{r}) =\frac{\mu_0}{4 \pi} \frac{3 \vec{r} ( \vec{r} \cdot \vec{m}) - r^2 \vec{m}}{r^5} [/itex]

    2. Relevant equations

    i want to derive the spherical expressions, with [itex] \vec{m} [/itex] parallel with [itex] z [/itex] axes

    [itex]
    x = r \cos \theta; y = r \sin \theta; z=z
    [/itex]


    3. The attempt at a solution

    I dunno what to do... i've tried to write [itex] B_r = \sqrt{(B_x)^2 + (B_y)^2} [/itex] ... but i fail....
    the solution should be:

    [itex]
    B_r = \frac{\mu_0}{4 \pi} \frac{2 \cos \theta}{r^3}
    [/itex]

    [itex]
    B_\theta = \frac{\mu_0}{4 \pi} \frac{\sin \theta}{r^3}
    [/itex]


    i dunno how to manage the scalar product, i feel really dumb and im sorry for the low quality of my question,

    just some general indication about what to do would be sufficient,

    thank u so much.

    mahblah.
     
  2. jcsd
  3. Aug 18, 2013 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    You should use spherical coordinates rather than cylindrical coordinates. See attached figure.

    Note that ##B_r = \vec{B}\cdot\hat{r}## and ##B_\theta = \vec{B}\cdot\hat{\theta}##

    So, you'll need to consider what you get from ##\vec{r}\cdot\hat{r}##, ##\vec{m}\cdot\hat{r}##, ##\vec{r}\cdot\hat{\theta}##, and ##\vec{m}\cdot\hat{\theta}##
     

    Attached Files:

  4. Aug 19, 2013 #3
    :)

    Thank u so much TSny!
     
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