State two applications of ampere�s law

Applications of Ampere law:-

  1. Infinite straight current-carrying wire:-

\(\begin{array}{l}\oint{\overrightarrow{B}\centerdot \overrightarrow{\delta}}={{mu }_{o}}I\end{array} \)

\(\begin{array}{l}B\oint{\overrightarrow{\delta }={{mu }_{o}}I},\Rightarrow B( 2\pi r)={{mu }_{o}}I\end{array} \)

\(\begin{array}{l}B=\frac{{{mu }_{o}}I}{2\pi r}\end{array} \)

\(\begin{array}{l}\Rightarrow \overrightarrow{B}=\frac{{{mu }_{o}}I}{2\pi r}\hat{\phi }\end{array} \)
(In cylindrical co-ordinates)

In this, the current is longitudinal (source is longitudinal) but the field produced is circumferential. The magnetic field is tangential to every point of the observed path.

  1. M.F of cylindrical current-carrying wire:-

A cylindrical wire having radius ‘R’, carrying a uniform current density

\(\begin{array}{l}\overrightarrow{J}\end{array} \)
. Once again the current here is longitudinal.

The magnetic field will be circumferential. We can choose a circular path as the integration path to calculate the magnetic field anywhere.

Inside:-

\(\begin{array}{l}\oint{\overrightarrow{B}}\cdot \overrightarrow{\delta}={{mu }_{o}}{{I}_{net}}\end{array} \)

\(\begin{array}{l}B\times 2\cancel{\pi }\cancel{r}=mu_{0}(J\cancel{pi }{{r}^{\cancel{2}}})\end{array} \)

\(\begin{array}{l}B=\frac{{{mu }_{o}}Jr}{2}\overrightarrow{B} (R)=\frac{{{mu }_{o}}(\overrightarrow{J}\times \overrightarrow{r})}{2}\end{array} \)

Outside:- 

\(\begin{array}{l}B=2\cancel{\pi }r={{mu }_{o}}J(\cancel{pi }{{R}^{2}})\end{array} \)

\(\begin{array}{l}B=\frac{{{mu }_{o}}J{{R}^{2}}}{2r}\overrightarrow{B}( R)=\frac{{{mu }_{o}}{{R}^{2}}}{2r}(\overrightarrow{J}\times \hat{r})\end{array} \)

For outside points, the M.F is similar to the field attained in the case of an st. current-carrying wire. That means, for outside points, it is behaving as if the entire current is passing through the axis.

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