Bio-Savart Law [Doubt about Cross Product in the equation]

Question

In my physics textbook, Bio-Savart Law is written as:

$$\vec{B} = \frac{K\,i\, d\vec{s} \times \vec{r}}{ 4 \pi \, r^2}$$

$K$: constant

And, when the cross-product is made, the result is:

$$B = \frac{K\,i\,ds\,\sin(\theta)}{4 \pi \, r^2}$$

My question is: shouldn't it be $r^3$ before the cross-product? I don't know why the "$r$" in the cross-product doesn't appear in the second equation

Thanks in advance.

Answer 1

It's an issue of normalizing the $\mathbf{r}$ vector. In the first equation your "r(v)" should really be $\hat{\mathbf{r}}$, the unit vector, written otherwise as $$\hat{\mathbf{r}}=\mathbf{r}/r.$$ In other words $d\mathbf{s}\times \mathbf{\hat{r}}=|d\mathbf{s}||\hat{\mathbf{r}}|\sin(\theta)=ds\sin(\theta)$.