91Ó°ÊÓ

Consider asphere of linear magnetic material is placed in an otherwise uniform magnetic field ${\mathrm{B}}_0$.

Write the formula of new magnetic field inside the sphere.

${\stackrel{\mathbf{→}}{\mathbf{B}}}_{\mathrm{∞}}={\stackrel{\mathbf{→}}{\mathbf{B}}}_0{\underset{\mathbf{n}=\mathbf{0}}{\overset{\mathrm{∞}}{∑}}\left(\frac{2}{3}\frac{{\mathrm{χ}}_m}{{\mathrm{χ}}_m+\mathbf{1}}\right)}^n$ …… (1)

Here,${\stackrel{\mathbf{→}}{\mathbf{B}}}_0$ is uniform magnetic field and${\mathrm{χ}}_m$ is magnetic susceptibility.

I'll apply the solution to issue 4.23. A magnetization is caused by the original magnetic field.

$\begin{array}{c}{\stackrel{→}{M}}_0={\mathrm{χ}}_m{\stackrel{→}{H}}_0\\ =\frac{{\mathrm{χ}}_m}{\mathrm{μ}}{\stackrel{→}{B}}_0\end{array}$

Determine the modifies magnetic field within the sphere:

$\begin{array}{c}{\stackrel{→}{B}}_1={\stackrel{→}{B}}_0+\frac{2}{3}{\mathrm{μ}}_0{\stackrel{→}{M}}_0\\ ={\stackrel{→}{B}}_0\left(1+\frac{2}{3}\frac{{\mathrm{χ}}_m}{{\mathrm{χ}}_m+1}\right)\end{array}$

Determine the magnetization induced by this field is:

$\begin{array}{c}{\stackrel{→}{M}}_1={\mathrm{χ}}_m{\stackrel{→}{H}}_1\\ =\frac{{\mathrm{χ}}_m}{\mathrm{μ}}{\stackrel{→}{B}}_1\\ =\frac{{\mathrm{χ}}_m}{\mathrm{μ}}{\stackrel{→}{B}}_0\left(1+\frac{2}{3}\frac{{\mathrm{χ}}_m}{{\mathrm{χ}}_m+1}\right)\end{array}$

Now, field is further modified:

$\begin{array}{c}{\stackrel{→}{B}}_1={\stackrel{→}{B}}_0+\frac{2}{3}{\mathrm{μ}}_0{\stackrel{→}{M}}_1\\ ={\stackrel{→}{B}}_0+\frac{2}{3}{\mathrm{μ}}_0\frac{{\mathrm{χ}}_m}{\mathrm{μ}}{\stackrel{→}{B}}_0\left(1+\frac{2}{3}\frac{{\mathrm{χ}}_m}{{\mathrm{χ}}_m+1}\right)\\ ={\stackrel{→}{B}}_0\left(1+\frac{2}{3}\frac{{\mathrm{χ}}_m}{{\mathrm{χ}}_m+1}+{\left(\frac{2}{3}\frac{{\mathrm{χ}}_m}{{\mathrm{χ}}_m+1}\right)}^2\right)\end{array}$

Where this song and dance will take us is fairly evident. The final magnetic field is obviously ${\stackrel{→}{B}}_{\mathrm{∞}}$:

Determine the new magnetic field inside the sphere.

$\begin{array}{c}{\stackrel{→}{B}}_{\mathrm{∞}}={\stackrel{→}{B}}_0\frac{1}{1−\frac{2}{3}\frac{{\mathrm{χ}}_m}{{\mathrm{χ}}_m+1}}\\ ={\stackrel{→}{B}}_0\frac{3({\mathrm{χ}}_m+1)}{3({\mathrm{χ}}_m+1)−2{\mathrm{χ}}_m}\\ =\frac{3{\mathrm{μ}}_r}{{\mathrm{μ}}_r+2}{\stackrel{→}{B}}_0\end{array}$

Therefore, the value of new magnetic field inside the sphere is ${\stackrel{→}{B}}_{\mathrm{∞}}=\frac{3{\mathrm{μ}}_r}{{\mathrm{μ}}_r+2}{\stackrel{→}{B}}_0$.