## A Dielectric Sphere in a Homogeneous Electric Field

In this problem we will encounter the main physical features of dielectric spheres - their induced field and polarizability. The result is of great interest: understanding the interaction of light with small particles is one of the main concerns of nanophotonics. It leads to astonishing applications like cloaking, improvement of solar cells and lithography with extreme resolution.

## Problem Statement

A dielectric sphere with radius $$r_{s}$$ and relative permittivity $$\varepsilon_{\mathrm{in}}$$ is embedded in a medium with relative permittivity $$\varepsilon_{\mathrm{out}}$$. The sphere is subject to a homogeneous electric field $$\mathbf{E}\left(\mathbf{r}\right)=E_{0}\mathbf{e}_{z}$$.

Calculate the electrostatic potential in the entire space. Derive the polarizability $$\alpha$$ of the sphere from your result.

## Background

The study of light-matter interactions is at the core of nanophotonics. Other than in classical optics, the investigated systems are only in the order of nanometers to a few microns. These systems are often made of basic building blocks like nanowires, rings with a gap (“split rings”), and spheres. Understanding these fundamental elements has lead to amazing new applications like the possibility to cloak certain regions from electromagnetic radiation.For example, in “Cloaking dielectric spherical objects by a shell of metallic nanoparticles”, Physical Review B 83 (2011), the authors show how a dielectric sphere can be cloaked surrounding it by silver spheres. The principle behind the paper is that the polarization of a dielectric sphere and that of the silver spheres can have different signs. Then, finding the right parameters of the system, the induced fields cancel each other - the dielectric sphere is invisible!