Question on an example relating to magnetic boundary condition.

[yungman]

This is an example shown in "Introduction to Electrodynamics" by Griffiths. Page 226 example 5.8.

Given a sheet of current K on the xy-plane where current traveling in +ve x direction. Find the magnetic field.

I am confused on the way the book justify the z direction of B is zero.

The book said:

Suppose the field pointed away from the plane. By reversing the direction of the current, I could make it point toward the plane ( in the Biot-Savart law, changing the sign of the current switches the sing of the field). But the z-component of B cannot possibly depend on the direction of the current in the xy-plane. Therefore there is no z component, the B only has y component.[End quote]

My question is why? Even the y component switch direction when the direction of the current is reversed. Can someone explain this.


Usually other books claimed the path length in z direction of surface current can be made to approach zero and therefore we let the z direction component equal zero for the case of surface current on xy-plane.

Thanks

 

[Bill_K]

Reverse the current, then rotate the plane through 180 degrees. Now the current is flowing in the same direction it was to begin with, yet somehow B_z is different. Only way to avoid a contradiction is if B_z = 0.

 

[yungman]

Reverse the current, then rotate the plane through 180 degrees. Now the current is flowing in the same direction it was to begin with, yet somehow B_z is different. Only way to avoid a contradiction is if B_z = 0.

In what axis you rotate the plane of current?

 

[hikaru1221]

The explanation of the book is quite too short and not so clear I think.
See the attached picture. I think this is the idea of the book:
_ For the left side of the picture, z-component of B field is in +z direction for upper half and -z for lower half, because of symmetry about the xy plane. The relative position of the current (red arrow) and the z-component of B field (blue arrows) is shown in the next figure.
_ For the right side of the picture, the current is flipped to -x direction, and thus, the z-component of B field is also flipped. The relative position is shown in the next figure.
_ Comparing the 2 relative position pictures, as B field must only depend on the direction of the current and not the axes, the z-component must be 0.

 

[AlephZero]

Rotate the plane 180 degrees around the z axis.

You should be able to see this just by symmetry. You have a sheet of current flowing in one direction in a plane. If there is a component of B perpendicular to the plane, then somehow the current must "know" which side of the plane is the "top" and which side is the "bottom", so it "knows" it is supposed to create the B vector pointing "up" instead of "down".

That makes no sense, so the component of B perpendicular to the plane must be 0.

 

[yungman]

thanks guys. I have not responded for a day because I am still a little unclear.

Let me confirm: So what all of you saying is it does not make sense if you rotate the current sheet so the direction of current reverse, the direction of the perpendicular B on both side of the Ampian loop reverse also. This cannot physically happened ( as if the B have intellenge to know the direction change). Therefore the perpendicular B must be zero. Is that the argument?

Thanks

Alan