PES 631

Problem 1.
Consider a molecule at the origin with axial symmetry about the axis having no net charge or dipole moment, but a quadrupoled moment . The molecule interacts with a charge at point .
Show that the force on the quadrupole is

(b) Show that the torque on the quadrupole is

Problem 2.

When a spherically symmetric atom is placed in an electric field, the charge distribution distorts to form a dipole moment. In the linear approximation the induced dipole moment is

where is the atomic polarizability.
(a) Show that relative to infinity (where the externally applied field is assumed to vanish) the energy of the atom in the electric field is

where is the energy density of the electric field and

is a volume that is typically of the order of the volume of the atom. Thus, neutral atoms are drawn into regions where the electric field is highest.
(b) The atom can be represented crudely by a nucleus surrounded by a rigid, uniform, spherical distribution of negative charge of radius , such that the nucleus can be pulled off center by the externally applied electric field against the restoring force of the spherical charge distribution. Compute the restoring electric field inside the negative charge distribution, and show that for this simple atomic model the polarizability is

Problem 3.

A dielectric sphere of radius a and dielectric constant ?1is placed in a dielectric liquid of infinite extent and dielectric constant ?2. A uniform electric field E was originally present in the liquid. Find the resultant electric field inside and outside the sphere.
, such that the nucleus can be pulled off center by the externally applied electric field against the restoring force of the spherical charge distribution. Compute the restoring electric field inside the negative charge distribution, and show that for this simple atomic model the polarizability is

Problem 3.

A dielectric sphere of radius a and…