Direction of Magnetic Field Question

[mmmboh]

In Griffith's page 226, there is an infinite sheet of current in the xy plane with a uniform surface current kî. He goes on to make arguments on why the magnetic field only points in the -y direction. I understand why there isn't a magnetic field in the x direction, but one of his reasons for there not being one in the z direction I don't understand. He says "Suppose the field pointed away from the plane. By reversing the direction of the current, I could make it point toward the plane. But the z-component of B cannot possible depend on the direction of current in the xy plane (think about it). " Well I've thought about it, and even asked a friend, and we don't understand this argument. Why couldn't the z-component of the magnetic field depend on the direction of current in the xy plane? if you change the direction of current, why shouldn't the field change directions in the z direction?

He has a similar example with a very long solenoid, and says that there is no radial field, because suppose B_s were positive, then if we reversed the direction of current B_s would now be negative, but switching the current direction is physically the same as turning it upside down, and that should not alter the radial field. But I don't understand why not, why wouldn't the radial field just switch directions?

Honestly the hardest part of doing magnetism for me is figuring out the directions of the field can someone enlighten me on this please?

 

[Delphi51]

I don't understand the "B cannot possible depend on the direction of current in the xy plane" either. As a counter-example, I propose a current circulating around the z axis in the x-y plane. This current causes a magnetic field along the z axis. You see this when dropping a magnet into a ring and the magnetic field due to the current it induces in the ring opposes the motion of the magnet, causing it to fall slowly. It is quite a dramatic demo if you drop a strong magnet through a copper tube.

A current in a wire along the x-axis causes a magnetic field to circulate around it. Say positive current flow. Then the magnetic field is in the z direction on one side of the wire, -z on the other. BUT with a current flowing in an infinite plane, like zillions of side-by-side wires going in the x direction you get a little B circulating around each and each one's up is canceled out by the next one's down. So no B field in the z direction. There would be at the edge of the current plane, but it is infinite so no edge.

 

[mmmboh]

Yeah he presents the argument you gave too, but he says the one I didn't understand is "nicer". What about for the solenoid, do you understand the argument in that case? I can come up with my own "reasons", but they might just be me trying to figure out what he means and being wrong.

 

[Delphi51]

The solenoid argument does make sense to me. Say the radial component of the field is in toward the center of the solenoid. Then if you reverse the current, the field should reverse and be outward. But if the observer stands on his head, the current appears to reverse . . .
It can't be both inward and outward so it must be zero.

 

[McCoy13]

This is old, but I just encountered this part of the text and it also confused me. The solenoid example cleared it up for me though.

Instead of flipping the direction of the current in the plane, think of rotating the plane 180 degrees. Then the current would be running the opposite direction. If the field were to change direction, then it would somehow have to depend on the orientation of our plane relative to our coordinate axis, meaning that the perpendicular component at a particularly distance from the plane might go from being +1 as we rotate the plane from \theta=0 to 0 at \theta=\frac{\pi}{2} to -1 at \theta=\pi, but our coordinates are totally arbitrary, so this is clearly preposterous.