Question

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 Ewould pbe proportional in that case? Find the condition on such that Eq. 4.1 will hold in the weak-field limit.

Short answer

The dipole moment of a charge density proportional to the distance from the centre is proportional to the square root of the electric field. For the electric field to be directly proportional to the dipole moment, the charge density has to be constant.

Step-by-step solution

Given data

The charge density of the electron cloud is

$\rho =Ar.....\left(1\right)$

Here, Ais the proportionality constant and r is the distance from the centre.

Electric field on a spherical Gaussian surface

The electric field on the a spherical Gaussian surface of radius ris

$\mathbf{E}\mathbf{=}\frac{{\mathbf{Q}}_{\mathbf{enc}}}{\mathbf{4}{\mathbf{\tau \tau \epsilon }}_{\mathbf{0}}{\mathbf{r}}^{\mathbf{2}}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$

Here, ${\mathrm{Q}}_{\mathrm{enc}}$is the charge enclosed by the Gaussian surface.

The infinitesimal volume element in spherical polar coordinates with just radial dependence is

$\mathbf{d\tau }\mathbf{=}\mathbf{4}{\mathbf{\tau \tau r}}^{\mathbf{2}}\mathbf{dr}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathbf{3}\mathbf{\right)}$

Derivation of dipole moment in terms of electric field

The expression for the charge enclosed is

$Q_{enc}={\int }_0^r\rho \left(r\text{'}\right)d\tau \text{'}$

Substitute the expression for the charge density from equation (1) and volume element from equation (3)

$$\begin{gathered} Q_{enc}=4\pi {\int }_0^{r}Ar\text{'}r{\text{'}}^{2}dr\text{'} \\ =\pi A{r}^4 \end{gathered}$$

Substitute expression for ${\mathrm{Q}}_{\mathrm{enc}}$in equation (2)

$$\begin{gathered} E=\frac{1}{4\pi {\epsilon }_0{r}^2}\pi A{r}^4 \\ =\frac{A{r}^2}{4{\epsilon }_0} \end{gathered}$$

Rearranging this

$r=\sqrt{\frac{4{\epsilon }_{0}E}{A}}$

The expression for the dipole moment is

$p=qr$

Substitute the expression for rand get

$p=q\sqrt{\frac{4\epsilon E}{A}}$

Thus, the dipole moment is proportional to $\sqrt{E}$.

For the electric field to be proportional to the dipole moment, the electric field has to be proportional to r. Thus, the charge density has to be constant.