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} =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 } .

=\oint (H _{\phi } \overrightarrow{a}_{\phi }) . \rho \ d\phi \overrightarrow{a}_{\phi }

=\int_{\phi = 0}^{2\pi } H _{\phi } \rho \ d\phi

= H _{\phi } \ \rho \ 2\pi .

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

H _{\phi } =\frac{ I}{\ 2\pi\ \rho } .

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

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

 

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Author: Lakshmi Prasanna Ponnala

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