Question

A sphere of linear magnetic material is placed in an otherwise uniform magnetic field ${\mathbf{B}}_0$. Find the new field inside the sphere.

Short answer

The value of new magnetic field inside the sphere is ${\overrightarrow{\mathrm{B}}}_{\mathrm{\infty }}=\frac{3{\mathrm{\mu }}_{\mathrm{r}}}{{\mathrm{\mu }}_{\mathrm{r}}+2}{\overrightarrow{\mathrm{B}}}_0$.

Step-by-step solution

Write the given data from the question.

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

Determine the formula of new magnetic field inside the sphere.

Write the formula of new magnetic field inside the sphere.

${\overrightarrow{\mathbf{B}}}_{\infty }={\overrightarrow{\mathbf{B}}}_0{\sum _{\mathbf{n}=\mathbf{0}}^{\infty }\left(\frac{2}{3}\frac{{\chi }_m}{{\chi }_m+\mathbf{1}}\right)}^n$ …… (1)

Here,${\overrightarrow{\mathbf{B}}}_0$ is uniform magnetic field and${\chi }_m$ is magnetic susceptibility.

Determine the value of new magnetic field inside the sphere.

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

$\begin{array}{c}{\overrightarrow{M}}_0={\chi }_m{\overrightarrow{H}}_0\\ =\frac{{\chi }_m}{\mu }{\overrightarrow{B}}_0\end{array}$

Determine the modifies magnetic field within the sphere:

$\begin{array}{c}{\overrightarrow{B}}_1={\overrightarrow{B}}_0+\frac{2}{3}{\mu }_0{\overrightarrow{M}}_0\\ ={\overrightarrow{B}}_0\left(1+\frac{2}{3}\frac{{\chi }_m}{{\chi }_m+1}\right)\end{array}$

Determine the magnetization induced by this field is:

$\begin{array}{c}{\overrightarrow{M}}_1={\chi }_m{\overrightarrow{H}}_1\\ =\frac{{\chi }_m}{\mu }{\overrightarrow{B}}_1\\ =\frac{{\chi }_m}{\mu }{\overrightarrow{B}}_0\left(1+\frac{2}{3}\frac{{\chi }_m}{{\chi }_m+1}\right)\end{array}$

Now, field is further modified:

$\begin{array}{c}{\overrightarrow{B}}_1={\overrightarrow{B}}_0+\frac{2}{3}{\mu }_0{\overrightarrow{M}}_1\\ ={\overrightarrow{B}}_0+\frac{2}{3}{\mu }_0\frac{{\chi }_m}{\mu }{\overrightarrow{B}}_0\left(1+\frac{2}{3}\frac{{\chi }_m}{{\chi }_m+1}\right)\\ ={\overrightarrow{B}}_0\left(1+\frac{2}{3}\frac{{\chi }_m}{{\chi }_m+1}+{\left(\frac{2}{3}\frac{{\chi }_m}{{\chi }_m+1}\right)}^2\right)\end{array}$

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

Determine the new magnetic field inside the sphere.

$\begin{array}{c}{\overrightarrow{B}}_{\infty }={\overrightarrow{B}}_0\frac{1}{1-\frac{2}{3}\frac{{\chi }_m}{{\chi }_m+1}}\\ ={\overrightarrow{B}}_0\frac{3({\chi }_m+1)}{3({\chi }_m+1)-2{\chi }_m}\\ =\frac{3{\mu }_r}{{\mu }_r+2}{\overrightarrow{B}}_0\end{array}$

Therefore, the value of new magnetic field inside the sphere is ${\overrightarrow{B}}_{\infty }=\frac{3{\mu }_r}{{\mu }_r+2}{\overrightarrow{B}}_0$.