Application of Ampere’s circuit law to infinite line current element

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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

The field $\overrightarrow{H}$ is either tangential (or) Normal to the path at each point of the closed path.
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 $\overrightarrow{H} =H _{\rho } \overrightarrow{a}_{\rho }+H _{\phi } \overrightarrow{a}_{\phi }+H _{z} \overrightarrow{a}_{z}$ .

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

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

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

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

$\oint \overrightarrow{H} . \overrightarrow{dl} = I$ .

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