Griffiths Example 5.8

Well he explains it later by saying that if you reversed the current, the B-field would point inward, which would be ridiculous.

EDIT: Never mind, I must have been thinking of a different example. No, he doesn't explain it. But the reason the Z-component of the B-field cannot depend on the direction of current is because reversing the current would give you ridiculous answers.

For example, if [tex]B_z[/tex] was uniform in the z-direction (up), reversing the current would make the B field point into the surface, which doesn't make sense.

If [tex]B_z[/tex] went like 1/r, then reversing the current would make it go like -1/r, which also doesn't make sense.

 

It is an infinite sheet of current. For any point, there will be as much current on one as the other. B_z due to current on one side would be up, but on the other down.

 

Yes, it does, but the point is that there is no magnetic field in the z-direction, because reversing the current would change the B-field direction.

Okay, think of it this way. It DOES have a component in the y-direction (current is in the x-direction, remember), so above the plate it is going +y, below the plate -y. Switch the current to go the opposite way, and above the plate it's -y and above it's +y. Everything still the same. If you look at it from a different angle, you can get back what you started with, right?

But if you have B going in the z-direction, then switching the current produces something completely different, not just a mirror, which doesn't make any sense at all because you're just switching the direction of current, which is totally arbitrary.