Problem 3P

According to Eq. 4.1, the induced dipole moment of an atom is proportional to the external field. This is a “rule of thumb,” not a fundamental law, and it is easy to concoct exceptions—in theory. Suppose, for example, the charge density of the electron cloud were proportional to the distance from the center, out to a radius R. To what power of E would p be proportional in that case? Find the condition on ρ(r ) such that Eq. 4.1 will hold in the weak-field limit.

Reference equation 4.1

p = αE.

Solution 3P

Step 1:

We need to find out the relation between electric field and the induced dipole.

From gauss law, the is electric field,

The enclosed charge, is,

And the charge density is,

is the area

Comparing this equation with equation ,