Definition and Explanation of Biot-Savart Law

Biot and Savart found experimentally that the magnitude of the field $dB$ at a point $P$, a distance $r$ from a small current element on a wire carrying current $I$, varies (i) directly with the current $I$; (ii) directly with the length $dl$ of the element; (iii) directly with $\sin\theta$, where $\theta$ is the angle between the element $d\vec l$ and $\hat r$ (the unit vector from the element toward $P$); and (iv) inversely with $r^2$. Combining these,

Here $d\vec B$ is perpendicular to both $d\vec l$ (the current direction) and $\hat r$. The net field at $P$ due to the whole conductor follows from superposition -- integrating $d\vec B$ over every current element:

$\vec B = \int d\vec B = \frac{\mu_0}{4\pi}\int \frac{I\,d\vec l \times \hat r}{r^2}$ …