# boundary condition for embedded dielectric sphere

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 $r=R$ ?

• What do you think and why? – sammy gerbil Dec 1 '16 at 2:20

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.