# application of Ampere’s circuit law to infinite line current element

Consider as infinitely long straight conductor placed along z-axis carrying a current I .

In order to determine $\overrightarrow{H}$ at some point P. we allow a closed path which passes through the point P and encloses the current  carrying conductor symmetrically such path is known as Amperian path.

To apply Ampere’s law the conditions t be satisfied are

1. The field $\overrightarrow{H}$ is either tangential (or) Normal to the path at each point of the closed path.
2. The magnitude of $\overrightarrow{H}$ must be same at all points of the path where  $\overrightarrow{H}$ is tangential.

Now,   $\overrightarrow{H}$  is given by  $\overrightarrow{H}&space;=H&space;_{\rho&space;}&space;\overrightarrow{a}_{\rho&space;}+H&space;_{\phi&space;}&space;\overrightarrow{a}_{\phi&space;}+H&space;_{z}&space;\overrightarrow{a}_{z}$ .

The path we are assuming is in the direction of $\phi$  so  $\overrightarrow{dl}&space;=&space;dl&space;\overrightarrow{a}_{\phi&space;}$ .

$\overrightarrow{dl}&space;=&space;\rho&space;\&space;d\phi&space;\overrightarrow{a}_{\phi&space;}$ .

Ampere’s law is used to find out $\overrightarrow{H}$  at P

i.e, from Ampere’s circuit law  $\oint&space;\overrightarrow{H}&space;.&space;\overrightarrow{dl}&space;=&space;I_{enc}$  .

$\oint&space;\overrightarrow{H}&space;.&space;\overrightarrow{dl}&space;=&space;I$ .

$\oint&space;\overrightarrow{H}&space;.&space;\overrightarrow{dl}&space;=\oint(H&space;_{\rho&space;}&space;\overrightarrow{a}_{\rho&space;}+H&space;_{\phi&space;}&space;\overrightarrow{a}_{\phi&space;}+H&space;_{z}&space;\overrightarrow{a}_{z})&space;.&space;\rho&space;\&space;d\phi&space;\overrightarrow{a}_{\phi&space;}$ .

$=\oint&space;(H&space;_{\phi&space;}&space;\overrightarrow{a}_{\phi&space;})&space;.&space;\rho&space;\&space;d\phi&space;\overrightarrow{a}_{\phi&space;}$

$=\int_{\phi&space;=&space;0}^{2\pi&space;}&space;H&space;_{\phi&space;}&space;\rho&space;\&space;d\phi$

$=&space;H&space;_{\phi&space;}&space;\&space;\rho&space;\&space;2\pi$ .

from Ampere’s law      $H&space;_{\phi&space;}&space;\&space;\rho&space;\&space;2\pi&space;=&space;I$ .

$H&space;_{\phi&space;}&space;=\frac{&space;I}{\&space;2\pi\&space;\rho&space;}$ .

$\therefore&space;\overrightarrow{H&space;_{\phi&space;}&space;}&space;=&space;\frac{&space;I}{\&space;2\pi\&space;\rho&space;}&space;\overrightarrow{a_{\phi&space;}}$ .

Ampere’s law is applied to find the value of $\overrightarrow{H}$  at any point P in it’s field.

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