Dipole of Magnetic field in polar coordinates

[mahblah]

Homework Statement

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}

Homework Equations

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

<br /> x = r \cos \theta; y = r \sin \theta; z=z<br />

The Attempt at a Solution

I don't know what to do... I've tried to write B_r = \sqrt{(B_x)^2 + (B_y)^2} ... but i fail...
the solution should be:

<br /> B_r = \frac{\mu_0}{4 \pi} \frac{2 \cos \theta}{r^3}<br />

<br /> B_\theta = \frac{\mu_0}{4 \pi} \frac{\sin \theta}{r^3}<br />


i don't know how to manage the scalar product, i feel really dumb and I am sorry for the low quality of my question,

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

thank u so much.

mahblah.

 

[TSny]

<br /> x = r \cos \theta; y = r \sin \theta; z=z<br />

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 $\vec{m}\cdot\hat{\theta}$

 

[mahblah]

:)

Thank u so much TSny!