91Ó°ÊÓ

Here, The sum of the enclosed currents times the permeability of free space is equal to the closed line integral of the magnetic field multiplied by the length of the curve, according to Ampere's law.

Write the expression for the ampere’s law.

$\mathbf{âˆ\circledR }\mathbf{B}\mathbf{â‹\dots }\mathbf{dl}\mathbf{=}{\mathbf{μ}}_{\mathbf{0}}∑I_{\mathrm{enc}}$ …… (1)

Here,${\mathrm{μ}}_0$ is the permeability for free space,$B$ is the magnetic field, $dl$is the length of curve, $I_{enc}$is the enclosed current.

Write the expression for value of current enclosed in terms of current density.

${\mathit{I}}_{\mathbf{e}\mathbf{n}\mathbf{c}}\mathbf{=}\underset{s}{∫}Jâ‹\dots da$ …… (2)

Here, $J$is the current density and$I_{enc}$ is the enclosed current.

The integral is surface independent, according to the divergence less field’s theorem. Any given boundary line's integral$∫Jâ‹\dots da$ value will be the same. For an enclosed surface, the integral value will be 0. In addition, the current density should have a lower divergence than the following criterion.

$∇â‹\dots J=0$ …… (3)

As a result, any particular surface can be considered for an endless number of surfaces with the same boundary line because the integral is independent of the surface.