How does the Lorentz Force relate to the fringing B field of a bar magnet?

[TFH]

Preface:
If a bar magnet is cut in half, we are left with say N-S--cut--N-S allinged along the z axis. Then the S and N ends are attracted to each other. This attraction is due to the "fringing" or non-uniform B field of the poles. That is the component of the B field that is normal to the "circular" magnetizing currents creates a Lorentz force along the Z axis.

So as the S side moves closer to N (the gap is reduced) the normal component of the B field increases.



Finally My questions:

(1) Why does the magnitude of the normal component increase. Is it because the magnitude of dB/dz is larger as the gap is closed? Or because the overall magnitude of B is larger and therefore it normal component is correspoundly larger? or ?


(2) When the Magnets touch, no air gap at all, shouldn't the force go to zero since the fringing or db/dz would go to zero or be greatly reduced? But I don't think this happens because my magnets stick to my Fridge...Help!

 

[kcdodd]

I think it would be due to the dB/dz getting larger, since that is the origin of the force. A dipole field drops off as 1/r^3, so derivative of that goes as 1/r^4. Another way to think about it is a minimization of energy in the field, since you're talking about a static situation. It takes less energy to be closer together.

 

[jrlaguna]

Energy minimization is a good way to see it: the direction of the force is always such that the energy is minimized most. The energy of the magnetic field is given by the density of lines (I can be more specific if you need). When the S pole of one magnet and the N pole of the other are nearby, the lines coming out of one die in the other very soon, so there are less B lines in space, so less energy.

 

[Antiphon]

This question and answer got off on the wrong track.

The attraction between north and south bar maget ends is *not* because of fringing fields.

In general, if a bar magnet is in a uniform magnetic field, it will experience a torque and not a net force becaue the attaraction and repulsion of the two poles cancel out. A dipole (which is what a small magnet is) in a non-uniform field will experince net forces because one ond of the magnetic dipole might be getting pushed or pulled more than the other.

But the N end of an infinitely wide bar magent with no fringing fields would be attracted to the S end of a similar infinitely wide "plane" bar magnet also with no fringing fields.

It is true that energy minimization leads to the correct answer but fringing fields are not necessary in this case. As the magnets get closer together, their fields are canceling each other out. Integrated over the space near the magnets, there is less energy in the magnetic field when they are together. (The energy goes out of the magnetic field and into your hands warming up your mucles.) This would still happen even without fringing. It also explains why the magnet hold on hardest when its together- the field is smallest but gets bigger very rapidly as you pull the magnet off the fridge.