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We all know about neutral points, which are nothing but points where the Earth's magnetic field neutralizes a magnet's magnetic field. When the north of a magnet faces geographic north, then the neutral points are perpendicular to the magnet's length. But I cannot understand why. For example, why don't the neutral points lie along the axis of the magnet? I've done a lot of research to try to determine the reason, but I could not find a answer. Would you please shed some light on the matter.

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The neutral (or critical) points of a vector field are the points where the field vector $\pmb{v}=\pmb{0}$.

For electromagnetic fields the superposition principle holds: if the field $\pmb{a}$ and field $\pmb{b}$ are solutions of the field equations, then $\pmb{a}+\pmb{b}$ also is. Hence, you can just add the dipole field from the bar magnet to the Earth's magnetic field and get the combination.

If the magnet is aligned with the field, then along its axis the vectors of the two component fields will line up and add to a somewhat stronger field, and there will not be any neutral points. But along the side of the magnet the field will be in opposite directions: the magnet is (roughly) a dipole and has a stronger field close and a weaker field far away: this means that at some distance from the equator of the magnet the two fields will be equal and opposite, and you get a neutral point. Or, to be strict, a neutral circle if it is a cylindrical magnet in 3D. If the magnet is instead aligned opposite to the field then the two fields will be parallel and reinforce each other along the equator, but there will be two points along the axis where they cancel each other.

An explicit way of calculating this is to use the formulas for the field from bar magnets and add a constant Earth field along the axis, seeing that they get zeros.

Field around a bar magnet directed opposite to a surrounding uniform magnetic field. Color denotes field strength. There are two neutral points along the magnet axis. (Field estimated by summing the contributions of a line of magnetic dipoles).

Field around a bar magnet directed along a surrounding uniform magnetic field. Color denotes field strength. There are neutral points outside the magnet's equator.

More abstract approach

In general, since magnetic fields are divergence-free ($\pmb{\nabla}\cdot\pmb{B}=0$) the possible isolated critical points cannot be sources or sinks: they have to be either 1:2 saddle points (field lines going in along one direction and out along two) or 2:1 saddle points (2 in directions, one out direction). They apparently cannot be spiral saddles either. This defines a skeleton of invariant manifolds where field lines connect the critical points (and infinity) along 1D lines or 2D sheets.

In the case with the non-aligned magnet one of the polar critical points is a 2:1 saddle point where the outgoing manifolds are lines connecting them to the nearest pole and infinity, while the incoming manifold is a ellipsoidal sheet emanating as the outgoing manifold from the other point, which is a 1:2 saddle point with incoming manifolds connecting it to the pole and infinity.

In the aligned magnet case the neutral points form a circle which acts as a 2D saddle point: there is an incoming sheet from infinity and an outgoing sheet going to one pole. This situation is not structurally stable: when you tilt the magnet the circle breaks up into two isolated saddle points that then move over to the non-aligned case as the magnet gets turned 180 degrees.

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