A sphere of linear dielectric material surrounded by another dieletric material

[ghostfolk]

Homework Statement

A sphere of linear dielectric material with permittivity $\epsilon_1$ and radius $a$ is surrounded by an infinite region of linear permittivity $\epsilon_2$. In the spherical region, there is free charge embedded given by $\rho_{free}=\beta r^2$, $0<r<a$, where $\beta$ is a constant and $r$ is the distance from the center of the sphere. Find the electric displacement $\vec{D}$ in all space.

Homework Equations

$\oint D \cdot da=Q_{free,enc}$

The Attempt at a Solution

Inside the sphere:
$\oint D \cdot da=4 \pi r^2$
$Q_{free,enc}=\begin{cases} \frac{4}{5} \beta \pi r^5 \hat{r}, r<a \\ \frac{4}{5} \beta \pi a^5 \hat{r}, r>a \end{cases}$
Here is where I am confused. I know that inside $\vec{D}=\frac{\beta}{5}r^3 \hat{r}$, but I'm not sure about how to find $\vec{D}$ outside. Normally $\vec{D}$ would be the same inside and out if the sphere were in empty space, but since it is now surrounded by a dieletric I am confused. Any help is appreciated.

 

[mark726]

I would first start by clarifying the problem statement. It seems that the problem is asking for the electric displacement $\vec{D}$ in all space, not just inside the sphere. This means that we need to find $\vec{D}$ both inside and outside the sphere.

To do this, we can use the boundary condition for electric displacement at the interface between two dielectric materials: $$D_{\perp,1}=D_{\perp,2}$$ where $D_{\perp,1}$ and $D_{\perp,2}$ are the components of $\vec{D}$ perpendicular to the interface in the two materials. In this problem, the interface is the surface of the sphere.

Inside the sphere, we can use the result we already have for $\vec{D}$: $\vec{D}=\frac{\beta}{5}r^3 \hat{r}$. Since the interface is at $r=a$, we have $D_{\perp,1}=\frac{\beta}{5}a^3$.

Outside the sphere, we can use the fact that there is no free charge in the infinite region. This means that $Q_{free,enc}=0$, and therefore $\oint D \cdot da=0$. This leads to $D_{\perp,2}=0$.

Now, using the boundary condition, we can equate $D_{\perp,1}$ and $D_{\perp,2}$ to get: $$\frac{\beta}{5}a^3=0$$ This leads to $\vec{D}=0$ for $r>a$.

Therefore, the final solution for $\vec{D}$ is: $$\vec{D}=\begin{cases}
\frac{\beta}{5}r^3 \hat{r}, r<a
\\ 0, r>a
\end{cases}$$ where $\beta$ is the constant given in the problem statement.