Q.(a) State Biot - Savart law and express it in the vector form.

Biot–Savart: $d\vec B=\dfrac{\mu_0}{4\pi}\dfrac{I\,d\vec l\times\hat r}{r^2}$; axial field of a coil $=\dfrac{\mu_0 I r^2}{2(r^2+x^2)^{3/2}}$.

1. Statement. The magnetic field $d\vec B$ due to a current element $I\,d\vec l$ at a point P at position $\vec r$ from the element has magnitude $dB=\dfrac{\mu_0}{4\pi}\dfrac{I\,dl\sin\theta}{r^2}$ ($\theta$ = angle between $d\vec l$ and $\vec r$), directed perpendicular to the plane of $d\vec l$ and $\vec r$. In vector form: $d\vec B=\frac{\mu_0}{4\pi}\,\frac{I\,d\vec l\times\hat r}{r^2}.$
2. Field on the axis of a circular coil. Consider a coil of radius $r$ carrying current $I$; P is on the axis at distance $x$ from the centre. Each element is at distance $\sqrt{r^2+x^2}$ from P, and $d\vec l\perp\vec r$, so $dB=\frac{\mu_0}{4\pi}\frac{I\,dl}{(r^2+x^2)}.$ …