Magnet Phenomenon: The divergent region of a ring magnet

[Ask Why!]

Magnet Phenomenon: The "divergent" region of a ring magnet

Here's a puzzler:
I have a NdFeB ring magnet, 1.75" outer diameter x 1.375" inner diameter x 0.25" thick, N40 grade. It is axially magnetized so that the north pole is toward one face and the south pole is toward the opposite face. The following image shows the well-known theoretical orientation of the magnetic field around the magnet, given that the north pole is on top:

SIDE VIEW

I have been in the lab determining the magnetic field orientation and approximate strength of the magnet using a linear Hall Effect sensor. The attached image shows the test configuration, with the orange arrow representing the dimension that was varied in the observance of the phenomenon. The image is not to scale.

If I move the sensor across the magnet, I get the expected readings: Saturation in one field orientation directly over the rim of the ring magnet, then a reversal of orientation in the open center of the magnet. BUT, if I take a reading over the center of the magnet from more than 0.5" away from the direct center along the cylindrical axis, the orientation of the magnetic field is the opposite of that observed over the center of the ring from within 0.5". I am having trouble explaining why this is the case.

The following leads me to strongly believe that this phenomenon is NOT DUE TO HYSTERESIS: The same effect is observable even if I allow the sensor to return to its quiescent voltage and then approach over the open center from a distance. The field is the reverse of the theoretical field indicated in the above image, until 0.5" from the center of the ring, at which point it reverses to the expected orientation.
The analogous effect is observable on the opposite side of the ring, that is, if we put south on top and kept the sensor in the same orientation.

Can anyone explain what is happening in what I have been calling the "divergent" region?
Please indicate if you are hypothesizing or applying accepted theory.

 

[mfb]

I get the expected readings: Saturation in one field orientation directly over the rim of the ring magnet, then a reversal of orientation in the open center of the magnet.

That is not in agreement with your image, where you have the same orientation in both cases.
Far away from your magnet, it looks like a dipole - along the axis, the orientation of the field is the same as the orientation directly above the magnetic parts.

 

[Ask Why!]

That is not in agreement with your image, where you have the same orientation in both cases.
Far away from your magnet, it looks like a dipole - along the axis, the orientation of the field is the same as the orientation directly above the magnetic parts.

Point taken. I have attached a more accurate illustration of the expected field.

 

[mfb]

That is not what I would expect.

For large distances, the field is always a dipole field if there is a dipole component (as there are no magnetic monopoles) - and your magnet has a dipole component.

 

[vanhees71]

You find some calculations of the field of permanent magnets in the very good book

A. Sommerfeld, Lectures on Theoretical Physics, Vol. III (Electrodynamics)

Of course, the main contribution of the multipole expansion not too close to such a magnet is the dipole contribution.

 

[Ask Why!]

That is not what I would expect.

For large distances, the field is always a dipole field if there is a dipole component (as there are no magnetic monopoles) - and your magnet has a dipole component.

After probing the ring more carefully, I found the field is better described by the attached modified image. The green arrows show the field further from the magnet. Note the areas directly above and below the hole where the yellow and green direction arrows are tip to tip and tail to tail.

mfb, is this more what you would expect?

 

[technician]

What field pattern do you get if you remove the permanent magnet well away from the area?

 

[mfb]

mfb, is this more what you would expect?

That is what I expected, indeed.

 

[Ask Why!]

Perhaps the real question I should ask is:
Why does the region near the magnet diverge from the dipole field?

 

[jtbell]

It's because what you have is not an ideal dipole, which has an infinitesimal size. Each infinitesimal piece of your magnet acts as an ideal dipole (I think, considering it as a continuous substance and ignoring variations at the atomic scale). Each piece is at a different location, hence at a different distance and direction from the point at which you measure the field. Also, the magnet may not be uniformly magnetized, in which case the different pieces would have different dipole moments.