91Ó°ÊÓ

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

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

$\stackrel{\mathbf{→}}{\mathbf{E}}=\frac{{\mathbf{Q}}_{\mathrm{inc}}}{\mathbf{4}\mathrm{Ï€}{\mathbf{r}}^2{\mathbf{Î\mu }}_0}$ …… (1)

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

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

$\stackrel{\mathbf{→}}{\mathbf{D}}={\mathbf{Î\mu }}_0\stackrel{\mathbf{→}}{\mathbf{E}}+\stackrel{\mathbf{→}}{\mathbf{P}}$ …… (2)

Here,${\mathbf{Î\mu }}_0$ is relative pemitivity,$\mathbf{E}$ is electric field and$\stackrel{\mathbf{→}}{\mathbf{P}}$ is polarization.

The bound surface and volume charge are

${\mathrm{σ}}_b=\stackrel{→}{P}â‹\dots \widehat{n}=\left\{\begin{array}{l}−k/a\text{ },r=a\\ k/b\text{ },r=b\end{array}\right\}$

$\begin{array}{c}{\mathrm{ÒÏ}}_b=−∇â‹\dots \stackrel{→}{P}\\ =−\frac{1}{r^2}\frac{∂}{∂r}\left(kr\right)\\ =−\frac{k}{r^2}\end{array}$

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 $4\mathrm{Ï€}kr$for $Q_{inc}$into equation (1).

$\begin{array}{c}\stackrel{→}{E}=−\frac{4\mathrm{Ï€}kr}{4\mathrm{Ï€}{r}^2{\mathrm{Î\mu }}_0}\\ =−\frac{k}{r{\mathrm{Î\mu }}_0}\widehat{r}\end{array}$

Therefore, the value of electrical field produces in all the bound charge is $\stackrel{→}{E}=−\frac{k}{\mathrm{γ}{\mathrm{Î\mu }}_0}\widehat{r}$.

Determine the value of $D$.

$\begin{array}{c}{âˆ\circledR }_S\stackrel{→}{D}â‹\dots d\stackrel{→}{S}=Q_{f,inc}\\ =0\\ \stackrel{→}{D}=0\end{array}$

Since there is no free charge anywhere.

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

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

$\begin{array}{c}0={\mathrm{Î\mu }}_0\stackrel{→}{E}+\stackrel{→}{P}\\ \stackrel{→}{E}=−\frac{\stackrel{→}{P}}{{\mathrm{Î\mu }}_0}\end{array}$

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

$\stackrel{→}{E}=−\frac{k}{{\mathrm{Î\mu }}_{0}r}\widehat{r}$

Therefore, the value of electrical field produces there no free charge anywhere is$\stackrel{→}{E}=−\frac{k}{{\mathrm{Î\mu }}_0\mathrm{γ}}\widehat{r}$ .