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Here's a problem from Electrodynamics by Griffiths, which I've solved but need more clarification in:

A hydrogen atom (with the Bohr radius of half an angstrom) is situated between two metal plates 1 mm apart, which are connected to opposite terminals of a 500 V battery. What fraction of the atomic radius does the separation distance d amount to, roughly? Estimate the voltage you would need with this apparatus to ionize the atom. [Use the value of α in Table 4.1. Moral: The displacements we’re talking about are minute, even on an atomic scale.]

So, finding d is easy if the field E is given in terms of the potential across the plates and the distance between them. For the second part, however, we must find the minimum field required to ionise the hydrogen atom - and somehow setting d = R yields the desired value of E. Why is it so?

Some might argue that we set d = R so as to find the minimum field, but why is this the minimum? From what is obvious, even beyond d = R, the H atom's nucleus and electron will be under each other's influence and will attract each other. What then, is special about d = R?

Note: d refers to the distance between positive and negative charges of the dipole created due to polarisation by the externally applied field.

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I haven't looked at this section of Griffiths, but it seems that you are asked to estimate $ E_{crit} $, the critical electrical field above which the atom is ionised. Or, in other words: $ E_{crit} $ is the minimum electric field which will ionise the atom, and any field greater than that will also obviously ionise the atom. I have two points to make.

First, I think you might be confusing minimum here with lower bound. Here you're just trying to find a rough estimate of $ E_{crit} $ using some handwavy physical argument (described in the next paragraph). That doesn't mean that the estimated value will be a lower bound on the actual $ E_{crit} $, necessarily. (Of course, maybe you can find an argument that does make your estimate a lower bound but until you do there's no reason to believe so.)

If this isn't the point of confusion, my second point would be about why you might expect $ d = R $ to be a plausible estimate. Take an approximate model of the hydrogen atom wherein applying the field simply displaces the whole electron cloud without deforming it, and displaces the nucleus the other way, producing the dipole moment. (I think this would be a first order approximation in a multipole expansion?) Then if $ d > R $, the electron cloud no longer overlaps with the nucleus, so you could expect ionisation. I think this is certainly a fine order-of-magnitude sort of calculation.

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