Physic Labs
Step 1 of 4: The differential Biot–Savart law
dB⃗=μ04π I dl⃗×r^r2d\vec B = \frac{\mu_0}{4\pi}\,\frac{I\,d\vec l \times \hat r}{r^2}
Analysis

Each current element I dl⃗I\,d\vec l produces, at a point located by vector r⃗\vec r, a differential field of magnitude dB=(μ0/4π) I dlsin⁡θ/r2dB = (\mu_0/4\pi)\,I\,dl\sin\theta/r^2, directed perpendicular to the plane containing dl⃗d\vec l and r⃗\vec r by the right-hand rule. This is the 'Coulomb form' of magnetism: contributions fall as 1/r21/r^2 and are weighted by sin⁡θ\sin\theta (an element parallel to r⃗\vec r contributes nothing).