# Ampere’s Circuit law

Ampere’s Circuit law states that the line integral of the tangential component of $\overrightarrow{H}$ around a closed path is the same as the net current (Ienc) enclosed by the path.

i.e, $\oint&space;\overrightarrow{H}&space;.\overrightarrow{dl}&space;=&space;I_{enclosed}$ .

This is similar to Gauss’s law and can be applied to determine $\overrightarrow{H}$ when the current distribution is symmetrical it’s a special case of Biot-savart’s law.

Proof:-

Consider a circular loop which encloses a current element . Let the current be in upward direction then the field is in anti- clock wise .

The current which is enclosed by the circular loop is of infinite length then  $\overrightarrow{H}$at  any point A is given by

$\overrightarrow{H}&space;=&space;\frac{I_{enc}}{2\pi&space;R}&space;\overrightarrow{a}_{\phi&space;}$ .

$\overrightarrow{H}.\overrightarrow{dl}&space;=&space;\frac{I_{enc}}{2\pi&space;R}&space;\overrightarrow{a}_{\phi&space;}.dl&space;\overrightarrow{a}_{\phi&space;}$ .

$\overrightarrow{H}.\overrightarrow{dl}&space;=&space;\frac{I_{enc}}{2\pi&space;R}.dl$.

$\overrightarrow{H}.\overrightarrow{dl}&space;=&space;\frac{I_{enc}}{2\pi&space;R}.R&space;d\phi$ .

$\overrightarrow{H}.\overrightarrow{dl}&space;=&space;\frac{I_{enc}}{2\pi&space;}d\phi$ .

$\oint&space;\overrightarrow{H}&space;.&space;\overrightarrow{dl}&space;=&space;\int_{\phi&space;=0}^{2\pi&space;}&space;\frac{I_{enc}}{2\pi&space;}d\phi$ .

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

which is known as the integral form of Ampere’s circuit law.

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