91Ó°ÊÓ

We will find the magnitude and direction of the magnitude of the magnetic field a distance of a below the current sheet. We will consider two wires and at equidistance on either side of the point where the magnetic field is being calculated. We can see that the z component of the field cancels, where the net magnetic field above the sheet points towards -x direction and the below the sheet points toward +x direction as shown in the figure. Now applying the Ampere’s law ($âˆ\circledR \hat{B}={\mathrm{μ}}_0{l}_{encl}$), using which a rectangular path as shown in the second figure. The magnetic field is parallel to the sheet, which means that it is perpendicular to the side with a length of 2a, thus $âˆ\circledR \hat{B}â‹\dots d\hat{l}=0$, again on the other hand we have:$âˆ\circledR \hat{B}â‹\dots dl=2BL$

Now, from the above equation since there is n conductor per unit length measured along x-ais, then the enclosed current is given by:

$I_{encl}=InL,$

Where, I is the current in each of the conductors. Hence:

$2BL={\mathrm{μ}}_{0}InL$

Or,

$B=\frac{1}{2}{\mathrm{μ}}_{0}In$

Now, by the symmetry, the magnitude of the magnetic field a distance of a above the current sheet field a distance of a below, but in the opposite direction that is:

$B=\frac{1}{2}{\mathrm{μ}}_{0}In$

Which is in the opposite direction.