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The Magnetic Field

Any current element $I d\Vec{\ell}$ contributes $d\Vec{B}$ to the magnetic field $\Vec{B}$ at a given point in space:

$$d\Vec{B} \; = \; k_M \; {I \, d\Vec{\ell} \times \hat{r} \over r^2}$$ (17.8)

where $\hat{r}$ is the unit vector in the direction of $\Vec{r}$, the vector distance from the current element to the point in space where the magnetic field is being evaluated. Eq. (8) is known as the LAW OF BIOT AND SAVART. It is still not perfectly transparent, I'm sure you will agree, but it beats Eq. (4)!

  

Figure:  The magnetic field $d\Vec{B}$ at position $\Vec{r}$ due to a current element $I d\Vec{\ell}$ at the origin.