Biot-Savart Law

Magnetic field contributed by a current element Idl at position vector r from the element. The cross product encodes both magnitude (sinθ) and direction. Integrate over a complete circuit for total B.

Statement

The magnetic field $d\vec{B}$ at a field point P due to a current element $I\,d\vec{l}$:

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

$\vec{r}$ is the position vector from the element to P; $\hat{r} = \vec{r}/r$.

Magnitude

$$dB = \frac{\mu_0}{4\pi} \frac{I\,dl\,\sin\theta}{r^2}$$

$\theta$ is the angle between $d\vec{l}$ and $\hat{r}$.

• $dB = 0$ when $\theta = 0$ or $\pi$: points on the line of the element feel no field.
• $dB$ is maximum when $\theta = 90°$.

Direction

$d\vec{B}$ is perpendicular to the plane containing $d\vec{l}$ and $\hat{r}$, given by the right-hand rule on $d\vec{l} \times \hat{r}$.

Comparison with Coulomb's Law

Both are inverse-square laws. The cross product is the essential structural difference.

Superposition

For a complete circuit:

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