Calculating the atomic polarizability of an atom

In summary, the electric field outside a uniformly charged sphere is the same as the electric field of a point charge which is equal to the whole charge of the sphere. However, the electric field inside of the sphere is not the same as the field of a point charge.
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Homework Statement


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Hi! I got stuck in studying the book "introduction to electrodynamics" written by griffith
I attached the related pictures above. The page is p.161-p.162.
It's concerned with calculating the atomic polarizability of an atom consisting of an nucleus surrounded by a uniformly charged spherical cloud. I agree with the idea under an electric field E[itex]^{\rightarrow}[/itex] the atom is polarized so that it pulls the nucleus of the atom apart from the uniformly charged spherical cloud. As you can see in p.162, it is said that the dipole moment of the polarized atom is just qd. I think that this is the same dipole moment of two charges q, -q(opposite charges) separated apart from each other with distance d (that is, -q can be thought to be positioned at the center of the spherical cloud). I think to calculate the dipole moment of the polarized atom consisting of spherical cloud and the nucleus the integral formula [itex]\int[/itex]r[itex]^{\rightarrow}[/itex]ρ(r)dτ must be used. However, I'm not sure if it has the same value with qd. Do you think it is the exact value of the formula or qd is just an approximation of the value of the formula?
As said in the p.162, the electric field to the plus charge q caused by the spherical cloud is [itex]\frac{qd}{4\pi\epsilon_{0}a^{3}}[/itex]. This is equal to [itex]\frac{q}{4\pi\epsilon_{0}}[/itex][itex]\frac{d}{a}[/itex][itex]^{3}[/itex][itex]\frac{1}{d^{2}}[/itex].
If I assume a is approximately d, then it is equal to [itex]\frac{q}{4\pi\epsilon_{0}d^{2}}[/itex]. So, It has the same effect with the spherical cloud replaced by point charge -q positioned at the center of the spherical cloud. From this, I expect that the point charge -q might be used to approximate it. However, even though it has the same effect in producing electric field to the point charge q, does it also have the same effect in making dipole moment? And furthermore, how am I sure that a is approximately equal to d??

I hope someone else answers my question.
Thank you for reading my question!

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The Attempt at a Solution

 
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  • #2
The external field of a uniformly charged sphere or spherical shell is COMPLETELY indistinguishable from that of a point charge at its center.
 
  • #3
What I know is that the electric field outside of a uniformly charged sphere is the same as the electric field of a point charge which is equal to the whole charge of the sphere. However, as you can see in the picture, d is less than the radius of the sphere a. And, unfortunately, inside of the sphere, I think they do not have the same value anymore...
So, I'd like to know if even though the two arrangements of charges have different electric fields inside the sphere, they have the same dipole moment...
 
  • #4
No, I cannot see any pictures. The attachments are said to be invalid.

Anyway, even a charged sphere overlaps with some other point charge, we still have the superposition principle. So we can consider them separately, and we can still replace the sphere with a point charge. Outside the sphere, the field of these two point charges will be the same as the original sphere and the original embedded point charge. I assume you only care about the external field of polarized atoms in this case.
 
  • #5


The calculation of atomic polarizability is a complex concept and requires a thorough understanding of electrodynamics and quantum mechanics. However, I will try to provide a simplified explanation to help you with your question.

First, let us define atomic polarizability. It is a measure of the ability of an atom to be polarized in an electric field. In other words, it is a measure of the change in the dipole moment of an atom when it is subjected to an external electric field.

In the case of an atom consisting of a nucleus and a uniformly charged spherical cloud, the dipole moment can be calculated using the integral formula \int\rho(\textbf{r})d\tau, where \rho(\textbf{r}) is the charge density and d\tau is the volume element. This integral takes into account the distribution of charge within the atom.

However, as you correctly pointed out, the dipole moment can also be approximated by qd, where q is the charge of the nucleus and d is the displacement of the nucleus from the center of the spherical cloud. This approximation is valid in the limit where the size of the atom is much smaller than the distance between the nucleus and the center of the spherical cloud.

So, to answer your question, qd is an approximation of the dipole moment calculated using the integral formula. It is not the exact value, but it can give a good estimate in certain cases.

Furthermore, the assumption that a is approximately equal to d is also valid in the same limit mentioned above. This is because in this case, the size of the atom is negligible compared to the distance between the nucleus and the center of the spherical cloud.

In conclusion, the use of qd as an approximation for the dipole moment and the assumption that a is approximately equal to d are valid in certain limits. However, for a more accurate calculation of atomic polarizability, the integral formula should be used. I hope this helps!
 

1. What is atomic polarizability?

Atomic polarizability is a measure of how easily an atom's electron cloud can be distorted by an external electric field. It is a fundamental property of an atom that affects its interactions with other atoms and molecules.

2. How is atomic polarizability calculated?

Atomic polarizability is calculated using quantum mechanical methods, such as density functional theory or Hartree-Fock theory. These methods use mathematical equations to describe the behavior of electrons in an atom and can provide accurate predictions of atomic polarizability.

3. What factors affect the atomic polarizability of an atom?

The atomic polarizability of an atom is affected by its electron configuration, size, and shape. Atoms with larger electron clouds or more loosely bound electrons tend to have higher polarizabilities.

4. How does atomic polarizability relate to chemical bonding?

Atomic polarizability plays a significant role in chemical bonding. It affects the strength and type of bonding between atoms, as well as the properties of molecules, such as their melting and boiling points. Molecules with higher polarizabilities tend to have stronger intermolecular forces.

5. Can atomic polarizability be measured experimentally?

Yes, atomic polarizability can be measured experimentally using techniques such as X-ray crystallography or spectroscopy. These methods involve studying the interactions of atoms with electromagnetic radiation to determine their polarizabilities. However, these measurements can be challenging to perform accurately and often require sophisticated equipment.

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