Contradiction in the formation of an electric dipole

[guv]

Homework Statement

Imagine a hydrogen atom isolated initially. Then a uniform electric field $E$ is turned on at the atom. This pulls electron in one direction pushs proton to the other direction. Forming a dipole as explained in most textbooks. In addition the stronger the external electric field, the greater the separation between electron and proton. However, when examining the set up in details, one can see the balance of the electron or the proton is between two electric forces
$$e E = \frac{ke^2}{d^2}$$
where $d$ is the separation of the electron from the proton. This balance equation predicts the distance must become smaller when external electric field $E$ becomes stronger, contradicting with what most books explain how induced dipoles form. Any suggestion on how to reconcile this? Thanks,

Relevant Equations

$$e E = \frac{ke^2}{d^2}$$
$$d = \sqrt{\frac{ke}{E}} \propto \sqrt{\frac{1}{E}}$$

Solution as stated in the problem description.

 

[Steve4Physics]

Hi.

Your analysis treats the proton and electron in a hydrogen atom as if they are classical, stationary, non-accelerating, point charges separated by a distance d. This is wrong and leads to totally incorrect results.

If we use the simple Bohr model, the electron moves in a circular orbit (radius d) with a particular angular momentum, so has (centripetal) acceleration which *must* be accounted for.

Thinking classically, the external electric field changes the shape of the orbit so that it becomes an ellipse with its semi-major axis longer than d and the proton at a focus, causing the atom to acquire a dipole moment.

A more rigorous quantum mechanical analysis is needed really, as the idea of point charges moving in definite orbits is wrong.

 

[rude man]

Homework Statement:: Imagine a hydrogen atom isolated initially. Then a uniform electric field $E$ is turned on at the atom. This pulls electron in one direction pushs proton to the other direction. Forming a dipole as explained in most textbooks. In addition the stronger the external electric field, the greater the separation between electron and proton. However, when examining the set up in details, one can see the balance of the electron or the proton is between two electric forces
$$e E = \frac{ke^2}{d^2}$$
where $d$ is the separation of the electron from the proton. This balance equation predicts the distance must become smaller when external electric field $E$ becomes stronger, contradicting with what most books explain how induced dipoles form. Any suggestion on how to reconcile this? Thanks,
Relevant Equations:: $$e E = \frac{ke^2}{d^2}$$
$$d = \sqrt{\frac{ke}{E}} \propto \sqrt{\frac{1}{E}}$$

Solution as stated in the problem description.

Don't forget that in the absence of an external E field the electron (classically) whizzes around the nucleus, so the coulomb force is offset by the centripetal force.

If an external E field is exerted on the atom, there will be a net attraction of the electron in the direction opposite the external field. The atom (electron & proton) forms a dipole moment in the same direction as that of the external field.

Edit: I edited out the equation of equilibrium since it's most likely wrong.
You'd have to take changes in orbital dynamics into account which I don't want to do.