Biot–Savart law

Remarks

For a steady current $I$ in a thin closed wire $\gamma$, the formula becomes the line integral $$B(x) = \frac{\mu_0 I}{4 \pi} \oint_\gamma \frac{d\ell \times (x - y)}{\abs{x - y}^3}.$$ In the sense of magnetostatics this is the definition of the field of a current, and it is the experimental law of Biot and Savart. The mathematical content is that $B = \nabla \times A$ with $A(x) = \frac{\mu_0}{4\pi} \int J(y) / \abs{x - y} \, dy$, so that $\nabla \cdot B = 0$ and $\nabla \times B = \mu_0 J$ when $\nabla \cdot J = 0$. Example: take the limit of an infinite straight wire along the third axis with current $I$. At distance $s$ from the wire, the formula gives a field of size $\mu_0 I / (2 \pi s)$, circling the wire. The hypothesis $\nabla \cdot J = 0$ is necessary for $\nabla \times B = \mu_0 J$, since the divergence of a curl is zero.

Going deeper

For a student

The Biot–Savart law plays for magnetostatics the role that Coulomb's law plays for electrostatics. From it one derives Gauss's law for magnetism and Ampère's circuital law, the two magnetostatic Maxwell equations. It is also used to compute the field of coils, for example the uniform field inside a long solenoid.