Determining the direction of the magnetic field of an infinite sheet of current

[LostInToronto]

This is a question about determining the direction of the magnetic field so we can use Ampere's law.

I'm working out of the 3rd edition of Griffiths' textbook. Ex. 5.8 on page 226 is "Find the magnetic field of an infinite uniform surface current $$\textbf{K} = K \widehat{x}$$ flowing over the xy plane."

First we want to determine the direction of B. It's clear that there is no component of the magnetic field that is in the x-direction, because the Biot-Savart law shows that the B-field is perpendicular to the direction of the current.

Griffiths says there is a nice argument for why there is no component in the z-direction. Suppose the field points away from the plane. By reversing the current, we can make it point towards the plane (again from the B-S law). "But the z-component of B cannot possibly depend on the direction of the current in the xy plane."

I don't understand why this last sentence is so. My guess is that it is because the current can flow in infinitely many directions on the plane, but the z-component of B can only point either up or down.

Any clarification would be wonderful!

 

[Berko]

Well, suppose B had a component in the positive z direction. Then, if we reversed the flow (negative x direction), B would suddenly point in the negative z direction.

But the definition of where positive x is is arbitrary. So, how can the physical effect (B field) depend on an arbitrary choice?

Thus, there must be no B field in the z direction.

 

[LostInToronto]

Thank you.