Question

Suppose the field inside a large piece of magnetic material is B₀, so that ${\mathit{H}}_{\mathbf{0}}\mathbf{=}\mathbf{(}\mathbf{1}\mathbf{/}{\mathbf{\mu }}_{\mathbf{0}}\mathbf{)}\mathbf{}{\mathit{B}}_{\mathbf{0}}\mathbf{-}\mathit{M}$, where M is a "frozen-in" magnetization.

(a) Now a small spherical cavity is hollowed out of the material (Fig. 6.21). Find the field at the center of the cavity, in terms of B₀ and M. Also find H at the center of the cavity, in terms of H₀ and M.

(b) Do the same for a long needle-shaped cavity running parallel to M.

(c) Do the same for a thin wafer-shaped cavity perpendicular to M.

Figure 6.21

Assume the cavities are small enough so M, B₀, and H₀ are essentially constant. Compare Prob. 4.16. [Hint: Carving out a cavity is the same as superimposing an object of the same shape but opposite magnetization.]

Short answer

(a)

The values of field at the center of the cavity, in terms of B₀ andis $\overrightarrow{B}={\overrightarrow{B}}_0-\frac{2}{3}{\mu }_0\overrightarrow{M}$.

The value of at the center of the cavity, in terms of H₀ and is $\overrightarrow{H}={\overrightarrow{H}}_0+\frac{\overrightarrow{M}}{3}$.

(b)

The value of field for long needle-shaped cavity perpendicular to M is $\overrightarrow{B}={\overrightarrow{B}}_0-{\mu }_0\overrightarrow{M}$.

The value of for long needle-shaped cavity perpendicular to M is $\overrightarrow{H}={\overrightarrow{H}}_0$.

(c) The value of for a thin wafer-shaped cavity perpendicular to M is $\overrightarrow{H}=\frac{1}{{\mu }_0}$.

Step-by-step solution

Write the given data from the question.

Suppose the field inside a large piece of magnetic material is B₀.

Consider a small spherical cavity is hollowed out of the material in (Fig. 6.21).

Assume the cavities are small enough so M, B₀, and H₀ are essentially constant.

Determine the formula of field at the center of the cavity, H at the center of the cavity, field for long needle-shaped cavity perpendicular to M and H for long needle-shaped cavity perpendicular to M.

Write the formula offield at the center of the cavity, in terms of .B₀

$\overrightarrow{\mathbf{B}}\mathbf{=}{\overrightarrow{\mathbf{B}}}_{\mathbf{0}}\mathbf{+}{\overrightarrow{\mathbf{B}}}_{\mathbf{s}}$ …… (1)

Here, ${\overrightarrow{\mathbf{B}}}_{\mathbf{0}}$is field inside a large piece of magnetic material and ${\overrightarrow{\mathbf{B}}}_{\mathbf{s}}$is field inside of a uniformly magnetized sphere.

Write the formula of at the center of the cavity, in terms of H₀ and M

$\overrightarrow{\mathbf{H}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{\mu }}_{\mathbf{0}}}\overrightarrow{\mathbf{B}}$ …… (2)

Here, ${\mathit{\mu }}_{\mathbf{0}}$is permeability and $\overrightarrow{\mathbf{B}}$ is magnetic field at the center of the cavity in terms of B₀.

Write the formula offield for long needle-shaped cavity perpendicular to M.

$\overrightarrow{\mathbf{B}}\mathbf{=}{\overrightarrow{\mathbf{B}}}_{\mathbf{0}}\mathbf{+}{\overrightarrow{\mathbf{B}}}_{\mathbf{s}}$ …… (3)

Here, role="math" localid="1657695281686" ${\overrightarrow{\mathbf{B}}}_{\mathbf{0}}$is field inside a large piece of magnetic material and ${\overrightarrow{\mathbf{B}}}_{\mathit{s}}$is field inside of a uniformly magnetized sphere.

Write the formula of H for long needle-shaped cavity perpendicular to M.

role="math" localid="1657695633493" $\overrightarrow{\mathbf{H}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{\mu }}_{\mathbf{0}}}\overrightarrow{\mathbf{B}}$ …… (4)

Here, ${\mathbf{\mu }}_{\mathbf{0}}$is permeability and $\overrightarrow{\mathbf{B}}$ is magnetic field at the center of the cavity in terms of B₀.

Write the formula of H for a thin wafer-shaped cavity perpendicular to M.

$\overrightarrow{\mathbf{H}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{\mu }}_{\mathbf{0}}}\overrightarrow{\mathbf{B}}$ …… (5)

Here, ${\mathit{\mu }}_{\mathbf{0}}$is permeability and $\overrightarrow{\mathbf{B}}$ is magnetic field at the center of the cavity in terms of B₀.

(a) Determine the value of field at the center of the cavity, in terms of B0 and M.

The field inside of a uniformly magnetized sphere with magnetization $-\overrightarrow{M}$is:

role="math" localid="1657696622237" ${\overrightarrow{B}}_s=-\frac{2}{3}{\mu }_0\overrightarrow{M}$

Determine the field at the center of the cavity, in terms of B₀ and.

Substitute $-\frac{2}{3}{\mu }_0\overrightarrow{M}$for ${\overrightarrow{B}}_s$into equation (1).

role="math" localid="1657696633523" $\overrightarrow{B}={\overrightarrow{B}}_0-\frac{2}{3}{\mu }_0\overrightarrow{M}$

Therefore, the values of field at the center of the cavity, in terms of B₀ and M is $\overrightarrow{B}={\overrightarrow{B}}_0-\frac{2}{3}{\mu }_0\overrightarrow{M}$.

Determine the H at the center of the cavity, in terms of H₀ and M.

Substitute ${\overrightarrow{B}}_0$for $\overrightarrow{B}$into equation (2).

$\begin{array}{rcl}\overrightarrow{H}& =& \frac{1}{{\mu }_0}{\overrightarrow{B}}_0-\frac{2}{3}\overrightarrow{M}\\ & =& \frac{1}{{\mu }_0}\left({\mu }_0\overrightarrow{M}+{\mu }_0{\overrightarrow{H}}_0\right)-\frac{2}{3}\overrightarrow{M}\\ & =& {\overrightarrow{H}}_0+\frac{\overrightarrow{M}}{3}\end{array}$

Therefore, the value of at the center of the cavity, in terms of H₀ and M is $\begin{array}{rcl}\overrightarrow{H}& =& {\overrightarrow{H}}_0+\frac{\overrightarrow{M}}{3}\end{array}$.

(b) Determine the value of value of field for long needle-shaped cavity perpendicular to M and H for long needle-shaped cavity perpendicular to M.

The needle cavity's induced field will resemble an endless cylinder with magnetization $-\overrightarrow{M}$in this scenario. It causes a field that is:

role="math" localid="1657697322029" ${\overrightarrow{B}}_s=-{\mu }_0\overrightarrow{M}$

Determine the field for long needle-shaped cavity perpendicular to $\overrightarrow{M}$.

Substitute $-{\mu }_0\overrightarrow{M}$for ${\overrightarrow{B}}_s$into equation (3).

$\overrightarrow{B}={\overrightarrow{B}}_0-{\mu }_0\overrightarrow{M}$

Determine the H for long needle-shaped cavity perpendicular to $\overrightarrow{M}$.

Substitute ${\overrightarrow{B}}_0-{\mu }_0\overrightarrow{M}$for $\overrightarrow{B}$into equation (4).

$\begin{array}{rcl}\overrightarrow{H}& =& \frac{1}{{\mu }_0}\left({\overrightarrow{B}}_0-{\mu }_0\overrightarrow{M}\right)\\ & =& \frac{1}{{\mu }_0}{\overrightarrow{B}}_0-\overrightarrow{M}\\ & =& {\overrightarrow{H}}_0\end{array}$

(c) Determine the value of field for a thin wafer-shaped cavity and H for a thin wafer-shaped cavity perpendicular to M.

The cavity will only generate a little magnetic field if the wafer is very thin, and the field will be:

Determine the field for a thin wafer-shaped cavity.

$\overrightarrow{B}={\overrightarrow{B}}_0$

Determine the H for a thin wafer-shaped cavity perpendicular to M.

$\overrightarrow{H}={\overrightarrow{H}}_0+\overrightarrow{M}$

Therefore, the value of H for a thin wafer-shaped cavity perpendicular to M is $\overrightarrow{H}=\frac{1}{{\mu }_0}$.