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.