MagSPHERE.nb

a uniformly magnetized Sphere

1 - Presentation

The magnetic field around a uniformly magnetized spherical ball  has direction shown below. The direction and the strength of the field varies considerably with the point of observation.

figure 1 . A 3D view of the magnetic field around the ball.

The next figure displays a cutaway of the ball.The z  axis is in the Magnetization direction. 
This schematic drawing is independant of the azimuthal angle ϕ  for reason of  axial symmetry. 

Figure 2 schematic of an  axial cutaway of the ball.

We are considering a sphere of 5 mm diamater made of Alnico (remanent magnetic field 1 tesla, i.e 10 000 gauss).
Its magnetic moment m is 65.4 gauss x cm^3 i.e: 50.1 ampere x cm^2.

Figure 3 Variation of the magnetic field (magnitude) along an axial line situated at 1 mm of the central axis.

Figure 4 Variation of the magnetic field (magnitude) on the surface of the sphere.

Figure 5 Variation of the magnetic field  direction on the surface of the sphere.

Figure 6   Graphe of the magnetic field lines (B)  on an axial plane.

2 -  Physico-Mathematical formulation

We consider a sphere of radius b with a uniform magnetization M of magnititude (Subscript[M, 0]) and parallel to the z axis, embedded in the vacuum.
A very good representation is given by a spherical ball of a permanent magnet material such as s Alnico, Ferrite or NdIB (Neodymium, Iron, Boron)  composite in the air. These materials have a very high remanent magnetization  respectively, let's say : 10000, 4000, 12000 gauss.
In this situation, the effect of the very small magnetic permeability of the air (≈10^-6) around  the ball is negligeable in respect of the fiefd generated by the ball material.

Magnetization is a local property of the matter that arises from the tiny dipoles of the atomic substances. Volume magnetization which represents the density of cooperative dipoles, has the same dimensional unit as the magnetic field. It can, therefore, be expressed in any relevant unit: gauss, telsa, ampere/meter, etc...

It can be shown that inside magnetic material the magnetic field is given by the relation (assuming a coherent system of magnetic units) :

B = M  + Hm                                   

This relation arises from the fundamenatl law of conservation of the magnetic field which states that at any point in the space, the magnetic field is simply the summation od the field generated by all the sources. We have :

B▪  This is the total magnetic field. It is usually called Induction or also Magnetic flux density (acronym B).
M▪  This is the intrinsic field generated by the substance. It is usually called Magnetization (acronym M) but also Magnetic Polarisation (acronym P). 

H▪  This is the field resulting from any magnetic sourced including the substance. In this case H dependd of the shape of the substance.
For instance for a long bar, at the middle of the bar, H  is negligible. But for shorten shape such as a cube or a sphere the effect is a reduction of the Intrinsic Field M. The ration of H/M is called the  Demagnetization Coefficient  (acronym 𝒟). For a cube or a sphere we have:
𝒟 = -(1/3)   
consequently 
H= -(1/3) M     and      B =    (2/3) M

Therefore for a spherical ball, the internal induction B is only 2/3 of the intrinsic magnetization so respectivelely for the 3 cited materials : 6666, 2667 and 8000 gauss. 

We use the amperian representation of the Magnetization. Then  M is generated by azimuthal surface Jϕ.
The vector potential A is :
a) inside the sphere   :    Aϕ(y)   = (1/3) M  y
b) outside the sphere :  Aϕ(y,z)  = (1/3) M  y  (b^3) / (rt^3)  
where y and z  are the radial and the axial coordinatesand rt the spherical distance of teh target point
                                rt =  ((((y^2) + (z^2)))^((1/2))) 

The Magnetic Field is a obtained by the Maxwel equation
                                 B = ▽ x Aϕ
The 2 components  Br and  Bz of the magnetic field vector B  are :
a) inside the sphere    for rt <  b
                                    Br =   0
                                    Bz =    (2/3) M
b) outside the sphere      for rt >= b
                                    Br =     M   (b^3)  / (rt^5) ( y  z )
                                    Bz =    M   (b^3)  /  (rt^5)  (2 (z^2) - (y^2)  )/3                   

 The magnitude of B at the surface Bm  gives the apparent strength of the ball magnet is :
 
                               Bm =     M   (b^3)  /  (rt^5)  (Sqrt[(((y z))^2)+((((2 (z^2) - (y^2)  )/3))^2)])   

and the B vector angle θ is

                           θ(radian) = ArcTang [((3 y z)/(2  (z^2) - (y^2)))]

The magnetic moment is :

                           m =   (4/3)  Pi   (b^3) M

The uniformly magnetized sphere has a very special meanning in the theory of magnetism. Indeed, this is the perfect representation of the magnetic dipole since the fields,  outside the sphere are strictly dipolar, even close to the sphere. There are no higher multipoles.