Dipole of Magnetic field in polar coordinates

1. Aug 18, 2013

mahblah

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:

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

2. Relevant equations

i want to derive the spherical expressions, with $\vec{m}$ parallel with $z$ axes

$x = r \cos \theta; y = r \sin \theta; z=z$

3. The attempt at a solution

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

$B_r = \frac{\mu_0}{4 \pi} \frac{2 \cos \theta}{r^3}$

$B_\theta = \frac{\mu_0}{4 \pi} \frac{\sin \theta}{r^3}$

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. Aug 18, 2013

TSny

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:

• SphericalCoordinates.png
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3. Aug 19, 2013

mahblah

:)

Thank u so much TSny!