content_image/371211.jpg content_image/371212.jpg content_image/371213.jpg content_image/371214.jpg content_image/371215.jpg
for the electric potential Φ: 

<img src='content_image/371211.jpg'>

where ℜ 3 is the euclidian 3-dimensional space and Σ is the surface of the particle. The derivatives are taken with respect to the normal vector n to the surface Σ. 

Due to the mismatch between the polarization inside and outside the object, electric charges accumulate at the interface Σ and create an electric potential which counteracts the uniform electric field stimulus. The solution of the above Laplace problem (3) is therefore formally given by 

The single layer charge distribution µ induced by the normalized electric field is a solution of the following BIE, obtained by inserting solution (4) in equations (3) 

<img src='content_image/371212.jpg'>

<img src='content_image/371213.jpg'>

Here the parameter λ = ( ε$_{1}$ − ε$_{0}$ ) / ( ε$_{1}$ + ε$_{0}$ ) isolates all the information regarding the dielectric properties for this problem. 

On using the linear response operator M that acts on the Hilbert space of integrable functions on the surface Σ, 

the integral equation (5) is written as 

<img src='content_image/371214.jpg'>

<img src='content_image/371215.jpg'>

The integral operator (6) is the electric field generated by the single layer charge distribution µ along the normal to the surface. It encodes the geometric information and has several interesting properties [13–15]. Its spectrum is discrete and it is not difficult to show that all of its eigenvalues are bounded by the [-1/2, 1/2] interval. Although non-symmetric, the operator (6) has real non-degenerate eigenvalues. The eigenvectors are biorthogonal, i. e., 

