Magnetostatics Anomalie: Infinite Surface Currents

[Void123]

I have been looking at a problem in magnetostatics about sheets of infinite surface current, positioned along the xy plane. Now clearly there is an Amperian loop parallel to the yz-plane. I understand why there would be no x component of the magnetic field, but I don't understand why there would be no z (vertical) component. According to Griffiths, they 'cancel out' or through a clever argument he uses: the direction of the current cannot affect the vertical component. Either way, I don't understand the logic at work here.

Would appreciate some feedback.

Thanks.

 

[Born2bwire]

The basic source of a magnetic field is a dipole. So a point source current is going to create a dipole magnetic field, whose field lines look like dees. So if the point source is at the origin pointing in the z direction, then the magnetic field in the x-y plane will point along the z direction too. Only for z =\= 0 will the field have an x and/or y component.

So conceptually you can visualize the current sheet as an infinite superposition of these point source currents. If we have two point sources located one behind the other, say:

^
|

^
|

Then the fields will be something like:

   ___
 /      \       
|       |     
|       |       
|       |     +          _____
 \_____ /               /      \       
                       |       |     
                       |       |       
                       |       |     
                        \_____/     

 =

  _____
 /      \       
|       |     
|       |       
|       |          
|       |     
|       |       
|       |     
 \_____/

The dees are offset slightly along the direction of the current (not in the normal direction asI have done above). This means that the components near the ends of the dees cancel each other out. If this occurs over an infinite sheet, the adjacent currents will cancel out all the z components. It is similar to why a ring of current's magnetic field along the ring's axis is only in the axial direction.