5.1 Calculating the Polarizability Tensor of a Solid Body

1. If a solid conductor is placed in an exterior homogeneous electrostatic field E, then the induced charge distribution ?( t) appears on its surface. Therefore the conductor acquires the dipole moment

(5.1)

where t _i is the ith coordinate of the radius vector t of the point t at the surface ? of the conductor. Since the equations of electrostatics are linear, there is a linear relation between P and E:

(5.2)

(summation over the repeated indices is understood), where V is the volume of the conductor, ? _e is the dielectric permittivity of the exterior medium, the matrix ? _ij is called the polarizability tensor. The dipole moment is interesting in many applications, especially in scattering theory (see Chapter 7).

A more general definition of the dipole moment is as follows. Let ? ₀ = -( E, x) be the potential of the exterior homogeneous field, ? = ? ₀ + u be the potential of the total field. If the obstacle is finite, then

(5.3)

We assume here that the obstacle is electroneutral, that is, its total charge is zero. The vector P is called the dipole moment induced on the obstacle by the exterior field E.

2. Let the obstacle be a homogeneous body with dielectric constant ?.

Put

(5.4)

The polarizability tensor is defined...