Question

In calculating the current enclosed by an Amperian loop, one must,in general, evaluate an integral of the form

${\mathit{I}}_{\mathbf{e}\mathbf{n}\mathbf{c}}\mathbf{=}{\int }_{s}J\cdot da$

The trouble is, there are infinitely many surfaces that share the same boundary line. Which one are we supposed to use?

Short answer

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.

Step-by-step solution

Define function

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.

$\mathbf{\oint }\mathbf{B}\mathbf{\cdot }\mathbf{dl}\mathbf{=}{\mathbf{\mu }}_{\mathbf{0}}\sum I_{\mathrm{enc}}$ …… (1)

Here,${\mu }_0$ 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.

${\mathit{I}}_{\mathbf{e}\mathbf{n}\mathbf{c}}\mathbf{=}\underset{s}{\int }J\cdot da$ …… (2)

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

Determine solution

The integral is surface independent, according to the divergence less field’s theorem. Any given boundary line's integral$\int J\cdot da$ 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.

$\nabla \cdot J=0$ …… (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.