Ampere’s Circuit law

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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 (I_enc) enclosed by the path.

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

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

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

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

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

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

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

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

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

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

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Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems. View all posts by Lakshmi Prasanna Ponnala

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