Magnetizing a sphere

Imagine you had an iron sphere and you wanted to magnetize the sphere in a particular direction of your choosing. We know from classical magnetostatics that this is possible by applying an external magnetic field ( $\mathbf{B}_{ext} = \mu_{o} H_{ext}$ , where $\mu_{o}$ is the magnetic permeability of free space) strong enough to rotate all of the magnetic moments (electron spins) to be parallel to the field. In our current model of Ferromagnetism, we can think about the a magnetic material as having magnetic domains in which the magnetization is polarized in several directions, but one direction within each domain. The boundaries separating the magnetic domains of different magnetic polarizations are unsurprisingly called magnetic domain walls. All that is to say that in order to polarize the magnet along a direction of your choice, there is understandably going to be some resistance, especially from magnetic domains that are magnetically polarized antiparallel to your external magnetic field. This resistance can be thought of as having two parts: 1) a ‘demagnetizing’ field, $H_{d}$ and 2) a magnetic permeability, $\mu$ .

For the latter, certain materials are more susceptible to magnetization from external magnetic fields than others, in the sense that the magnetic moments within the magnet’s volume rotate more easily. This is described mathematically as the magnetic susceptibility, $\chi = \frac{\mu}{\mu_{o}}-1$ , where $\mu$ is the magnetic permeability of the material. $\chi$ varies with $H_{ext}$ , and is the slope of a simple linear relationship between the magnetization of a material per unit volume ( $I$ ) and the effective applied external magnetic field ( $H_{eff} = H_{ext} - H_{d}$ ), $I = \chi H_{eff}$ . This formulation of $H_{eff}$ helps to visualize what $H_{d}$ is: a magnetic field generated from the magnetic moments oriented in directions with an antiparallel component to the external field, $H_{ext}$ . $H_{d}$ adds in superposition with $H_{ext}$ , decreasing the strength of the magnetizing field. In addition to the magnetic moments, $H_{d}$ depends on the a demagnetization factor $N$ , which is dependent on the geometry of the magnet in question. I think of this factor as a way to account for magnetic moments’ tendency to align in a particular direction based on geometry, AKA shape-anisotropy. For an in-depth explanation of this factor, I’ll refer you to any textbook on Ferromagnetism, especially Chikazumi’s “Physics of Ferromagnetism.” For our purposes, $N_{sphere} = \frac{1}{3}$ , and $H_{d} = N\frac{I}{\mu_{o}}$ .

We’ve now developed the tools to study the initial question of magnetizing a magnetic sphere. When we magnetize a magnetic sphere with radius $R$ , the final magnetic moment of is proportional to the magnetization per unit volume ( $I$ ) produced by $H_{eff}$ : $M = V_{sphere} I$ . Let’s solve for the magnetization produced by $H_{eff}$ .

$I = \chi H_{eff} =\chi \left( H_{ext} - H_{d} \right) = \chi \left( H_{ext} - N \ frac{I}{\mu_{o}} \right)$
$I = \chi \left( H_{ext} - N \ frac{I}{\mu_{o}} \right) \Rightarrow I = \frac{\chi H_{ext}}{1+\frac{N \chi}{\mu_{o}}}$

This leads us to our final relationship:

$M = V_{sphere} I = \frac{4}{3}\pi R^{3} \frac{\chi H_{ext}}{1+\frac{N \chi}{\mu_{o}}}$

This is an interesting relationship for several reasons. Let’s make some plots in python that may help us visualize this result.


import numpy as np
import matplotlib.pyplot as plt

def M(vol, chi, H_ext, N_demag):
    '''Calculates the magnetic moment induced by an external magnetic field
    'H_ext' within a magnetic material with magnetic susceptibility 'chi'
    and a sample shape with volume 'vol' and demagnetization factor 'N_demag.'
    '''
    import numpy as np
    mu_o = 4*np.pi(10**(-7))
    return(vol*(chi*H_ext)/(1+(N_demag*chi/mu_o)))

MU_O = 4<em>np.pi</em>(10**(-7)) #H/m
N_PIX = 100
N_DEMAG = 1./3
R = np.linspace(0,1,N_PIX) #m
H_EXT = 1. #A/m
CHI = 1.
VOL = 4./3<em>np.pi</em>R**(3) #m^3 for a sphere with radius R

FTSIZE = 35
plt.figure(figsize=(10,7))
plt.scatter(R,M(vol=VOL, chi=CHI, H_ext = H_EXT, N_demag = N_DEMAG))
plt.xlabel("$R$ [m]", fontsize = 0.75*FTSIZE)
plt.ylabel("$\frac{M}{\chi\ H_{ext}}$ [Tm/A]", fontsize = 0.75*FTSIZE)
plt.xticks(fontsize = 0.6*FTSIZE)
plt.yticks(fontsize = 0.6*FTSIZE)
plt.tight_layout()
plt.savefig('M_vs_R.png',dpi=200)
plt.show()

I’ve plotted the reduced magnetic moment versus the radius of the circle. As you can see, it grows exponentially, but of course there is only so big you can make these magnetic spheres, so there is a practical limit to the magnetic moment. What about if we change the material to something with a higher magnetic susceptibility $\chi$ ?

N_PIX = 100
N_DEMAG = 1./3
R = 10**(-2) #m
H_EXT = 1. #A/m
CHI = np.linspace(0,100*MU_O,N_PIX)
VOL = 4./3*np.pi*R**(3) #m^3 for a sphere with radius R
FTSIZE = 35
plt.figure(figsize=(10,7))
plt.scatter(CHI,M(vol=VOL, chi=CHI, H_ext = H_EXT, N_demag = N_DEMAG))
plt.xlabel("$\chi$", fontsize = 0.75*FTSIZE)
plt.ylabel("$\frac{M}{H_{ext}}$ [Tm/A]", fontsize = 0.75*FTSIZE)
plt.xticks(fontsize = 0.6*FTSIZE)
plt.yticks(fontsize = 0.6*FTSIZE)
plt.ticklabel_format(axis="x", style="sci", scilimits=(0,0))
plt.tight_layout()
plt.savefig('M_vs_CHI.png',dpi=200)
plt.show()

Figure 2 shows that the reduced magnetic moment asymptotically approaches some value, which we can calculate by taking the limit $\lim_{\chi \to +\infty}$ $M(\chi) = \frac{4}{3}\pi R^{3} \frac{\mu_{o} H_{ext}}{1/3} = 4 \pi R^{3} H_{ext} \mu_{o}$ . For $R = 1 \textrm{[cm]}$ and $H_{ext} = 1 \textrm{[A/m]}$ , $M(\chi) \approx 1.58 \times 10^{-11}$ , which is consistent with Fig. 2. We can interpret this as the magnetic moment’s saturation value for a material’s intrinsic susceptibility. In other words, a magnet that is infinitely susceptible to magnetization will have a magnetic moment strength approaching this value.