(Solved): explanation step by step Stokes' Theorem: The magnetic field due to a current density J is (He ...

explanation step by step

Stokes' Theorem: The magnetic field due to a current density $\vec{J}$ is (Here ${\mu }_0$ is the permeability of the vacuum and $j_0$ and $a$ are constants of suitable dimension) $\vec{B}(\vec{r})=\{\begin{array}{ll}\frac{{\mu }_0{j}_0}{2}\left(x{\hat{e}}_y-y{\hat{e}}_x\right)& \text{ for }0\le \rho \le a\\ \frac{{\mu }_0{j}_0{a}^2}{2{\rho }^2}\left(x{\hat{e}}_y-y{\hat{e}}_x\right)& \text{ for }\rho \ge a\end{array}$ Compute the circulation of the magnetic field over a circle of radius $R$ lying in the $xy$-plane centred at the origin, in two different ways: (a) First, explicitly carry out the line integral. (b) Next, evaluate the integral using Stokes' Theorem. (c) Noting that $\vec{\nabla }\times \vec{B}={\mu }_0\vec{J}$, show that the circulation of $\vec{B}$ equals ${\mu }_0$ times the current through the planar surface enclosed by the circle. This is known as Ampere's Law.