biot savart question..

transgalactic said:

here is a question:
https://i27.tinypic.com/14tsro1.gif

a thin ring with uniform resistance [latex]\rho[/latex]
with radius [latex]r_0[/latex] connected to current I.
the ring is divided into two areas.
calculate the total magnetic field in the center


by current divider:
[latex]I_1=I\frac{l_2}{l_1+l_2}\\[/latex]
[latex]I_2=I\frac{l_1}{l_1+l_2}\\[/latex]
by biot savart law :
[latex]dB=\frac{\mu _0 I ds\times \hat{r}}{4\pi r^2}\\[/latex]
[latex]B_1=\int_{0}^{l_1}\frac{\mu _0 I ds }{4\pi r^2}(+\hat{k})\\[/latex]
[latex]B_2=\int_{0}^{l_2}\frac{\mu _0 I ds }{4\pi r^2}(-\hat{k})[/latex]
[latex]B=B_1+B_2[/latex]

and this is the formal solution
https://i29.tinypic.com/2moc8ds.jpg
they have r^3 in the denominator
why??
the crosst prodct of ds with r is 1
why they take cross product of theta with r??
i cant understand the solution
how they built this integral
??

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