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Is the potential across the boundary continuous for a dielectric sphere embedded in a dielectric material, so that the potential inside the sphere can be set equal to the potential outside of it at $r=R$ ?

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  • $\begingroup$ What do you think and why? $\endgroup$ – sammy gerbil Dec 1 '16 at 2:20
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In most problems it suffices to think about the boundary conditions on the $D$ field. At an interface between two dielectrics we have that (in cgs units) $$ \mathbf{n}\cdot\mathbf{D} |_+ - \mathbf{n}\cdot\mathbf{D} |_- = 4\pi \sigma $$ Where $\sigma$ is the surface charge density on the boudnary. Hence, $$ \epsilon_1 \mathbf{n}\cdot \nabla\phi|_+ - \epsilon_2 \mathbf{n}\cdot \nabla\phi|_- = -4\pi \sigma $$ where $\epsilon_1$ is the dielectric constant of the material on the outside of the boundary and $\epsilon_2$ is the dielectric constant of the material inside of the boundary.

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