I rather stuck on this homework question. I have been working on it for 3 days. I need a little help.
Context
A point dipole p is placed at the centre of a dielectric sphere with permittivity $\varepsilon$ and radius $R$.
A set of axes are chosen with the origin at the centre of the sphere and the dipole aligned along the z-axis.
The overall question is to find the electrostatic potential inside and outside of the sphere. I am stuck on a sub question.
Hint. In the usual notation, pose:
$$\Phi_{in}(r,\theta)=\frac{p \cos(\theta)}{4\pi \varepsilon r^2}+\sum_{l=0}^{\infty}A_{l}r^{l}P_{l}\cos(\theta)$$ in $r<R$
$$\Phi_{out}(r,\theta)=\sum_{l=0}^{\infty}\frac{B_{l}}{r^{l+1}} P_{l}\cos(\theta)$$ in $r>R$
where $p=|\textbf{p}|$. Use the boundary conditions to determine the $A_{l}$,$B_{l}$.
Question:
The boundary conditions here are:
$$(\textbf{D}_{out}-\textbf{D}_{in})\cdot \hat{\textbf{r}}=\sigma$$ $$\sigma=0 \color{blue}{ \text{ across }}r=R$$ $$\Rightarrow(\textbf{D}_{out}-\textbf{D}_{in})\cdot \hat{\textbf{r}}=0$$
$$(\textbf{E}_{out}-\textbf{E}_{in})\times \hat{\textbf{r}}=\textbf{0}\color{blue}{ \text{ across }}r=R$$
Show that these lead to here:
$$\lim_{r\rightarrow R^{+}}(\varepsilon_{0}\frac{\partial \Phi_{out}}{\partial r})=\lim_{r\rightarrow R^{-}}(\varepsilon\frac{\partial \Phi_{in}}{\partial r})$$
Notes
$\textbf{D}$ is called the electric displacement $$\textbf{D}=\varepsilon \textbf{E}$$
