Question

2. Consider again a system with charges +q and -q, but where the dipole moment is variable. Such a system is called polarizable, with a polarizability ? defined by p = ? E. Recall the classical model of an atom from last week: a cloud of uniform positive charge with radius R and total charge +q, and a point electron of charge -q.

a. What is the polarizability of this model atom?

b. What is its potential energy in a field of strength E? Assume for simplicity that the atom is in the configuration of lowest energy, with the dipole moment aligned with the external field. This is trickier than it looks. It helps to recall that this system behaves a spring, so you could compare your result to the result for an equivalent spring to check your answer.

c. Instead of uniform external field, suppose that there is a point charge +Q interacting with our polarizable model atom from a distance r. Find the induced dipole moment to leading order in r.

d. Find the potential energy of this system, and the (attractive) force between the point charge and the neutral atom (!)

e. It turns out that even two neutral atoms like these have an attractive electric interaction. This works because if one temporarily has a small dipole moment, it will produce an electric field which will polarize the other, and vice versa. Assuming this picture is correct, how should the interaction energy between two neutral atoms depend on the distance r? Note: you won't be able to derive this behavior from the model we're using; this problem is asking for qualitative reasoning based on the idea given.

Answer 1

Considering the system of two charges +q and -q as a classical model of an atom with uniform charge density of the electron cloud upto the atomic radius R.

a) Let us consider the atom is placed in an Electric Field E. The central positive charge is displaced by the effect of E to a distance 'a' along the direction of E. The nucleus is displaced by a force F= q E. The outer electron cloud will exert an electrostatic force on the nucleus to restore its displacement. By Gauss' Law, the part of electron cloud within the sphere of radius 'a' can only responsible for the restoring electrostatic force, which is given by

The atomic dipole p is given by

b)The potential energy of the dipole in an electric field E is

W = -p.E = p E cos \theta, \theta = 0 since dipole is aligned with E.

, which is similar to the potential energy stored in a spring.