Question

A thick spherical shell (inner radius a, outer radius b) is made of dielectric material with a "frozen-in" polarization

$\mathbf{P}\left(\mathbf{r}\right)=\frac{k}{r}\widehat{\mathbf{r}}$

Where a constant and is the distance from the center (Fig. 4.18). (There is no free charge in the problem.) Find the electric field in all three regions by two different methods:

Figure 4.18

(a) Locate all the bound charge, and use Gauss's law (Eq. 2.13) to calculate the field it produces.

(b) Use Eq. 4.23 to find $\mathbf{D}$, and then get$\mathbf{E}$ from Eq. 4.21. [Notice that the second method is much faster, and it avoids any explicit reference to the bound charges.]

Short answer

(a) The value of electrical field produces in all the bound charge is $\overrightarrow{E}=-\frac{k}{\gamma {\epsilon }_0}\widehat{r}$.

(b)

The value of$D$ is $0$.

The value of electrical field produces there no free charge anywhere is$\overrightarrow{E}=-\frac{k}{{\epsilon }_0\gamma }\widehat{r}$ .

Step-by-step solution

Write the given data from the question.

Consider a thick spherical shell (inner radius a, outer radius b) is made of dielectric material

Determine the formula of electrical field produces in all the bound charge and electrical field produces there no free charge anywhere.

Write the formula ofelectrical field produces in all the bound charge.

$\overrightarrow{\mathbf{E}}=\frac{{\mathbf{Q}}_{\mathrm{inc}}}{\mathbf{4}\pi {\mathbf{r}}^2{\mathbf{\epsilon }}_0}$ …… (1)

Here,${\mathbf{Q}}_{\mathrm{inc}}$ is charge inside of the sphere,$\mathbf{r}$ is radius of the sphere and${\mathbf{\epsilon }}_0$ is relative pemitivity.

Write the formula ofelectrical field produces there no free charge anywhere.

$\overrightarrow{\mathbf{D}}={\mathbf{\epsilon }}_0\overrightarrow{\mathbf{E}}+\overrightarrow{\mathbf{P}}$ …… (2)

Here,${\mathbf{\epsilon }}_0$ is relative pemitivity,$\mathbf{E}$ is electric field and$\overrightarrow{\mathbf{P}}$ is polarization.

(a) Determine the value of electrical field produces in all the bound charge

The bound surface and volume charge are

${\sigma }_b=\overrightarrow{P}\cdot \widehat{n}=\left\{\begin{array}{l}-k/a,r=a\\ k/b,r=b\end{array}\right\}$

$\begin{array}{c}{\rho }_b=-\nabla \cdot \overrightarrow{P}\\ =-\frac{1}{r^2}\frac{\partial }{\partial r}\left(kr\right)\\ =-\frac{k}{r^2}\end{array}$

Inside of the sphere $Q_{inc}=0$so the electric field is obviously zero. Now, in the middle region.

role="math" localid="1657547262465"

Determine the electric fieldproduces in all the bound charge.

Substitute $4\pi kr$for $Q_{inc}$into equation (1).

$\begin{array}{c}\overrightarrow{E}=-\frac{4\pi kr}{4\pi r^2{\epsilon }_0}\\ =-\frac{k}{r{\epsilon }_0}\widehat{r}\end{array}$

Therefore, the value of electrical field produces in all the bound charge is $\overrightarrow{E}=-\frac{k}{\gamma {\epsilon }_0}\widehat{r}$.

(b) Determine the value of electrical field produces there no free charge anywhere.

Determine the value of $D$.

$\begin{array}{c}{\oint }_S\overrightarrow{D}\cdot d\overrightarrow{S}=Q_{f,inc}\\ =0\\ \overrightarrow{D}=0\end{array}$

Since there is no free charge anywhere.

Now, determine the electrical field produces there no free charge anywhere.

Substitute$0$ for$\overrightarrow{D}$ into equation (2).

$\begin{array}{c}0={\epsilon }_0\overrightarrow{E}+\overrightarrow{P}\\ \overrightarrow{E}=-\frac{\overrightarrow{P}}{{\epsilon }_0}\end{array}$

This shows that the electric field is zero between outside and inside ( $r>b$and $r<a$, respectively) and between:

$\overrightarrow{E}=-\frac{k}{{\epsilon }_{0}r}\widehat{r}$

Therefore, the value of electrical field produces there no free charge anywhere is$\overrightarrow{E}=-\frac{k}{{\epsilon }_0\gamma }\widehat{r}$ .