# Confusion regarding Ampere's law

Consider the case in which we have two parallel sheets on the $$x$$-$$y$$ plane with a distance $$d$$ between them. Their orientation is along the $$x$$ direction. Each sheet has a surface current $$I$$, where the current on the above sheet is in the $$+x$$ direction and the current on the below sheet is in the $$-x$$ direction. If we need to find the magnetic field between these two sheets, we will construct a closed contour on the above sheet and calculate the field as follows according to Ampere's law:

$$\int (B.dl)= \mu I_{enclosed}$$

The closed contour constructed will be a rectangle of approximately zero thickness to enclose the sheet of zero thickness. There will be two sides only of the contour in the positive and negative $$y$$ directions each with length $$w$$. Therefore the integration will be: $$B \omega + B\omega = \mu I$$ $$B=\mu I/2\omega$$ The same will be done to the other sheet and the magnetic field between the two sheets will be the same of the contributions from each one. My question is: Why did we consider the four sides of the closed contour while calculating the field for each sheet for the the region between the sheets? Why is it not possible to consider only the side that exists in the region between the sheets and Ampere's law will be: $$B\omega=I \mu$$??

Moreover, if we need to calculate the field in the region above the two plates it will be zero, since each plate will contribute a magnetic field in opposite directions to each other in the required region. Is this right?

• Please, consider adding a diagram that shows your situations. Thank you. Apr 9, 2019 at 20:41

$$\oint \vec{B}\cdot d\vec{l}=\mu_0 I_{enc},$$
Why the closed contour appears can be tracked back to looking at Maxwell's equations: $$\vec{\nabla}\times \vec{B}=\mu_0\vec{J}$$. To get Ampere's law from this, integrate over a closed contour and apply Stoke's theorem. If you don't integrate over a closed contour, the result does not hold.