State two applications of ampere�s law

Applications of Ampere law:-

  1. Infinite straight current carrying wire:-
\(oint{overrightarrow{B}centerdot overrightarrow{dell }}={{mu }_{o}}I\) \(Boint{overrightarrow{dell }={{mu }_{o}}I},,Rightarrow ,Bleft( 2pi r right)={{mu }_{o}}I\) \(B=frac{{{mu }_{o}}I}{2pi r}\) \(Rightarrow ,overrightarrow{B}=frac{{{mu }_{o}}I}{2pi r}hat{phi }\) (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 an uniform current density\(overrightarrow{J}\) . 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:- \(oint{overrightarrow{B}}cdot overrightarrow{dell }={{mu }_{o}}{{I}_{net}}\) \(Btimes 2cancel{pi }cancel{r}=mu o(Jcancel{pi }{{r}^{cancel{2}}})\) \(B=frac{{{mu }_{o}}Jr}{2},,,,,overrightarrow{B}left( rle R right)=frac{{{mu }_{o}}(overrightarrow{J}times overrightarrow{r})}{2}\)

Outside:- \(B=2cancel{pi }r={{mu }_{o}}J(cancel{pi }{{R}^{2}})\) \(B=frac{{{mu }_{o}}J{{R}^{2}}}{2r},,,,,overrightarrow{B}(rge R)=frac{{{mu }_{o}}{{R}^{2}}}{2r}(overrightarrow{J}times hat{r})\) 

For outside points, the M.F is similar to the field attained in case of a 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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