91Ó°ÊÓ

Reference as Ex. 4.2

Consider $\stackrel{→}{P}$will be point along the z-axis.

Consider$I$ will be using as letter.

Consider$l$ as the distance from the source to the point of interest.

Write the formula of potential of auniformly polarized sphere.

$\mathbf{V}\left(\stackrel{\mathbf{→}}{\mathbf{r}}\right)=\frac{1}{\mathbf{4}\mathrm{Ï€}{\mathbf{Î\mu }}_0}{∫}_{\mathrm{ν}}\frac{\stackrel{\mathbf{→}}{\mathbf{P}}\left({\mathbf{r}}^{\mathbf{\text{'}}}\right)â‹\dots \widehat{\mathbf{I}}}{{\mathbf{l}}^2}\mathbf{d}{\mathrm{Ï„}}^{\text{'}}$ …… (1)

Here, $\stackrel{\mathbf{→}}{\mathbf{p}}$is vector constant in both magnitude and direction,$\stackrel{}{\mathbf{}\mathbf{}\mathbf{r}}$ is inner radius of sphere,$\widehat{\mathbf{I}}$will be using as letter, $\mathit{l}$as the distance from the source to the point of interest and role="math" localid="1657544577909" ${\mathbf{Î\mu }}_0$is relative pemitivity.

Write the formula ofpolarization vector for the electric field of a homogenous sphere inside the sphere.

$\mathbf{V}(\mathbf{r}\mathbf{,}\mathbf{θ})=\stackrel{\mathbf{→}}{\mathbf{P}}â‹\dots \frac{\stackrel{→}{r}}{\mathbf{3}{\mathbf{Î\mu }}_0}$ …… (2)

Here, $\stackrel{\mathbf{→}}{\mathbf{P}}$is vector constant in both magnitude and direction, $\stackrel{\mathbf{→}}{\mathbf{r}}$is radius of sphere and role="math" localid="1657544779642" ${\mathbf{Î\mu }}_0$is relative pemitivity.

Write the formula of polarization vector for the electric field of a homogenous sphere outside the sphere.

$\mathbf{V}(\mathbf{r}\mathbf{,}\mathbf{θ})=\stackrel{\mathbf{→}}{\mathbf{P}}â‹\dots \frac{{\mathbf{R}}^3}{\mathbf{3}{\mathbf{Î\mu }}_0{\mathbf{r}}^2}\widehat{\mathbf{r}}$ …… (3)

Here, $\stackrel{\mathbf{→}}{\mathbf{P}}$is vector constant in both magnitude and direction, $\stackrel{\mathbf{→}}{\mathbf{r}}$is radius of sphere, $\mathbf{R}$is outer radius of sphere and ${\mathbf{Î\mu }}_0$is relative permittivity.

The distance $I$between the source and the place of interest will be represented by the letter $I$due to site limitations.

The sphere has continuous polarization (so $\stackrel{→}{P}$is a vector constant in both magnitude and direction). Specify $\stackrel{→}{P}$as the z-axis pointer. The potential using (eq. 4.9) is:

Determine the potential of a uniformly polarized sphere.

Substitute$1$for${\stackrel{→}{r}}^{\text{'}}$into equation (1).

$V\left(\stackrel{→}{r}\right)=\stackrel{→}{P}â‹\dots \frac{1}{4\mathrm{Ï€}{\mathrm{Î\mu }}_0}{∫}_{\mathrm{ν}}\frac{\hat{l}}{l^2}d{\mathrm{Ï„}}^{\text{'}}$

The electric field of a homogeneous sphere of charge with $\mathrm{ÒÏ}=1$may be calculated accurately by multiplying the polarization vector. So, for $r<R$:

Determine thepolarization vector for the electric field of a homogenous sphere of charge inside the sphere.

Substitute$r\cos \mathrm{θ}$for $\stackrel{→}{r}$into equation (2).

$V_{ins}(r,\mathrm{θ})=\frac{\stackrel{→}{P}r\cos \mathrm{θ}}{3{\mathrm{Î\mu }}_0}$

Therefore, the value of polarization vector for the electric field of a homogenous sphere of charge inside the sphere $\mathrm{ÒÏ}=1$is $V(r,\mathrm{θ})=\frac{\mathrm{Prcos}\mathrm{θ}}{3{\mathrm{Î\mu }}_0}$.

Determine thepolarization vector for the electric field of a homogenous sphere of charge outside the sphere.

Substitute$\cos \mathrm{θ}$ for $\hat{r}$into equation (3).

$V_{out}(r,\mathrm{θ})=\frac{\stackrel{→}{P}{R}^3\cos \mathrm{θ}}{3{\mathrm{Î\mu }}_0{r}^2}$

Therefore, the value of polarization vector for the electric field of a homogenous sphere of charge outside the sphere $\mathrm{ÒÏ}=1$is $V(r,\mathrm{θ})=\frac{\mathrm{P}{R}^3\cos \mathrm{θ}}{3{\mathrm{Î\mu }}_0{r}^2}$.