# Calculating the atomic polarizability of an atom

## 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$^{\rightarrow}$ 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 $\int$r$^{\rightarrow}$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 $\frac{qd}{4\pi\epsilon_{0}a^{3}}$. This is equal to $\frac{q}{4\pi\epsilon_{0}}$$\frac{d}{a}$$^{3}$$\frac{1}{d^{2}}$.
If I assume a is approximately d, then it is equal to $\frac{q}{4\pi\epsilon_{0}d^{2}}$. 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!!

## The Attempt at a Solution

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The external field of a uniformly charged sphere or spherical shell is COMPLETELY indistinguishable from that of a point charge at its center.

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...

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.