Question

A point charge $$q$$ is located on the place dividing vacuum and infinite uniform isotropic dielectric with permittivity $$\epsilon$$. Find the moduli of the vectors $$D$$ and $$E$$ as well as the potential $$\varphi$$ as functions of distance $$r$$ from the charge $$q$$.

Answer 1

$$E_{p} = \dfrac {q\vec {r_{1}}}{4\pi \epsilon_{0}r_{1}^{3}} + \dfrac {q'\vec {r_{2}}}{4\pi r_{2}^{3}\epsilon_{0}}, P$$ in $$1$$

$$E_{p} = \dfrac {q''\vec {r_{1}}}{4\pi \epsilon_{0} r_{1}^{3}}, P$$ in $$2$$

where $$q'' = \dfrac {2q}{\epsilon + 1}, q' = q'' - q$$

In the limit $$\vec {l}\rightarrow 0$$

$$\vec {E}_{p} = \dfrac {(q + q')\vec {r}}{4\pi \epsilon_{0} r^{3}} = \dfrac {q\vec {r}}{2\pi \epsilon_{0} (1 + \epsilon) r^{3}}$$, in either part.

Thus, $$E_{p} = \dfrac {q}{2\pi \epsilon_{0} (1 + \epsilon)r^{2}}$$

$$\varphi = \dfrac {q}{2\pi \epsilon_{0} (1 + \epsilon) r}$$

$$D = \dfrac {q}{2\pi \epsilon_{0}(1 + \epsilon)r^{2}}$$

$$\times \{\begin{matrix}1\ in\ vacuum\\ \epsilon\ in\ dielectric

\end{matrix}\right.$$