Induced dipole moment due to application of an external electric field to an atom

david20

Hi all, I am having a hard time understanding the following scenario (I'm quoting from page 305 of the "Electricity & Magnetism" textbook which you can access via Google Books):

"Assume a simple model for an atom which consists of a point nucleus (+q) surrounded by a uniformly charged spherical cloud (-q) ... in the presence of an external electric field E, the nucleus will be shifted slightly to the right and the electron cloud to the left." The text goes on to explain that "equilibrium occurs when the nucleus is displaced a distance d from the centre of the sphere. At that point the external field pushing the nucleus to the right exactly balances the internal field pulling it to the left."

On the surface that makes sense, but what is vexing me is how we get from the initial condition to the final condition. As the +q and -q move away from each other, the electrostatic force that attracts them will oppose the force due to the external electric field. However, the further they move apart, the weaker that force. If the two forces balance each other perfectly when they are a distance d apart, what accounts for the motion before then? The implication I am drawing is that before they were a distance d apart, the force due to the electric field trumped the attractive force. But if that is the case, then there's no way the forces would balance out at any distance b/c as distance increases the attractive force decreases. On the other side of the coin, if the force due to the external electric field was weaker than the attractive force then why did the charges move apart at all?

I'm making a faulty assumption somewhere here, but I have spent a few hours trapped in this horrible thought cycle so I thought I would reach out for help here! Thank you to those who take the time to help me point out where I went wrong!

monish

If you are dealing with two point charges, it is true that the force gets weaker as they are pulled apart. But when you analyze a point charge versus a distributed charge, the force gets stronger as they are pulled apart.

Ulysees

Actually with the model of the uniform charge cloud given, the force gets weaker as they are pulled apart (decreasing linearly to the surface of the cloud, and then as an inverse square, these are mathematical theorems for uniform spheres).

So why the force gets stronger as they are pulled apart, is not trivial at all. The force should weaken after a certain distance for sure, this much is obvious.

Ulysees

Ignore the above, it's incorrect, sorry. The force increases linearly to the surface and then falls as an inverse square, with a uniform charge distribution.

Ulysees

So here is the diagram, of force versus distance for a uniform sphere (of charge or mass).



Something similar must be happening with the actual charge distribution, increase then decrease.

david20

Hi all, thanks for your feedback. So the short story is that b/c the electron cloud has a uniform charge distribution there's a different set of dynamics at play. As the nucleus and electron cloud move apart, the force holding them together strengthens. Eventually this force will become strong enough to neutralize the force due to the external electric field. Phew! Thanks!