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The Biot-Savart law is the magnetic analog of Coulomb's law: $d\vec{B} = \frac{\mu_0}{4\pi}\frac{Id\vec{l} \times \hat{r}}{r^2}$, giving the field contribution from an infinitesimal current element $Id\vec{l}$ at a point displaced by $\vec{r}$. The cross product ensures the field is perpendicular to both the current element and the position vector, distinguishing it from the radial electric field of a point charge.

Key properties: (1) $dB \propto I$ (linear in current), (2) $dB \propto 1/r^2$ (inverse-square), (3) $dB = 0$ along the axis of the current element ($\sin\theta = 0$), and (4) $dB$ is maximum in the plane perpendicular to the element ($\sin\theta = 1$). The total field requires integration over the entire conductor.

Standard results derived from Biot-Savart: infinite straight wire ($B = \mu_0 I/(2\pi d)$), finite wire ($B = \frac{\mu_0 I}{4\pi d}(\sin\alpha + \sin\beta)$), semi-infinite wire ($B = \mu_0 I/(4\pi d)$), circular loop center ($B = \mu_0 I/(2R)$), and loop axis ($B = \mu_0 IR^2/(2(R^2+x^2)^{3/2})$). Each formula has specific geometric conditions that must be carefully identified in problems.