Ferromagnetism, what's happening in cylindrical magnets?

[soljiang]

I've a question on ferromagnetism. I've been trying to simulating the field around a rod shape magnet, cylindrical in shape and N on one end and S on the other. I just wonder if one would expect the the B, the magnetic flux density, to be uniform in magnet. I used comsol to simulate and get different results for rods with different aspect radio.
Basically the simulation demonstrates that the higher the aspect ratio (length/diameter), the more uniform the B is in the magnets. When the length is 2mm and the diameter is 6mm (aspect ratio 1/3), then the B is very strong at the periphery of the cylinder and weaker in the middle along the central axis.
I'd really appreciate if someone could offer me some help on mathematical and physical explanation, such as physical laws and maths formula. Thank you very much!


Charles

 

[soljiang]

Here's a fourth picture that shows the uniformity of the B when aspect ratio is high.