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 \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.


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: vikramarka

Completed M.Tech in Digital Electronics and Communication Systems and currently working as a faculty.