How Does Ampere's Law Explain Magnetic Fields at the Poles?

[cscott]

A north pole and south pole are separated by some distance (positioned vertically).

Using the discrete version of Ampere's law:

Take a path rectangle with one vertical side completely inside the magnetic field and the other vertical side completely outside the magnetic field.

Where h = height:
$$Bh = -\left[(B_{||}\Delta l)_{top} + (B_{||}\Delta l)_{bottom}\right]$$

So there is a magnetic field outside the line of the magnets' edge because the RHS doesn't equal 0? Why is the RHS negative?

 

[berkeman]

Since there's no such thing as a magnetic monopole, I assume that you mean there is a bar magnet at the origin with its north pole pointing up and the south pole pointing down. What does it mean when you say "completely outside the magnetic field"? That would be out at infinity?

 

[cscott]

Since there's no such thing as a magnetic monopole, I assume that you mean there is a bar magnet at the origin with its north pole pointing up and the south pole pointing down. What does it mean when you say "completely outside the magnetic field"? That would be out at infinity?

I meant the north end of a bar magnet pointing down with another bar magnet some distance below it with it's south pole pointing upwards.

I guess by outside of the magnetic field they mean away from the edge of the magnets and the fringe effect. The way I described my rectangular path is the hint they gave in the textbook. "Bh" is the vertical side inside the field, and the term for the other vertical side disappears because B = 0.

 

[berkeman]

I meant the north end of a bar magnet pointing down with another bar magnet some distance below it with it's south pole pointing upwards.

I guess by outside of the magnetic field they mean away from the edge of the magnets and the fringe effect. The way I described my rectangular path is the hint they gave in the textbook. "Bh" is the vertical side inside the field, and the term for the other vertical side disappears because B = 0.

So there are two bar magnets in series. Even so, the B field does not go to zero for that system except at infinity or with ideal magnetic field shielding. Something is missing here...

BTW, the "fringe effect" comes into play when you are working with calculating the capacitance of a finite size capacitor, not some magnetic geometry, IMO.