Effect of material size on magnetization

[Liam89]

I'm a bit confused about the effect of the size of ferromagnetic material on the magnetic field it generates so I was hoping someone could explain it to me.

The saturation magnetization is given by
M = $\frac{N}{V}$*μ
but this suggests that the magnetization is independant of the size of the material, but since
B = μ₀(H + M)
and since B fields add up vectorially this would suggest cutting a piece of ferromagnetic material in two would double the magnetic field, which doesn't make sense, so I was wondering what I was missing?

 

[Q-reeus]

What makes sense is that magnetized material can be thought of as composed of elementary magnetic dipoles m packed together with uniform volume density, so the number N_m of such dipoles goes as r³ - the cube of linear dimensions of a sample of characteristic length r. You know I take it that the field strength of each dipole m drops off with distance as 1/r³. So guess what one finds when combining those two factors.

 

[Per Oni]

I'm a bit confused about the effect of the size of ferromagnetic material on the magnetic field it generates so I was hoping someone could explain it to me.

The saturation magnetization is given by
M = $\frac{N}{V}$*μ
but this suggests that the magnetization is independant of the size of the material, but since
B = μ₀(H + M)
and since B fields add up vectorially this would suggest cutting a piece of ferromagnetic material in two would double the magnetic field, which doesn't make sense, so I was wondering what I was missing?

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/solenoid.html

The formula shows that for a long (L >> R) electromagnet: B=μ0 N/L I.

Perhaps I can use this formula for your case and say that if the current remains the same but N increases, what will happen? As long as L also increases B will remain the same.
So now if I then separate this long electromagnet into 2 (still long) electromagnets, with the currents still the same value, how much is B for each separate coil? Well it’s the same as before.

An interesting thing is: what happens to the flux when I separate? Answer: since: φ = B x A, it looks like if we separate these coils far enough we have twice the amount of flux. How much is the total magnetic energy of the coils before and after separation? Answer: the same.

Question: what happened to the energy I needed to separate the coils?