Estimate the voltage you need with this appartus to ionize the atom

[stunner5000pt]

A hydrogen atom (with Bohr radius of half an angstrom) is situated between 2 metal places 1 mm apart which are connected topoosite terminals of a 500 V battery.
What fraction of the atomic radius does the separation distance d amount to, roughly?

Estimate the voltage you need with this appartus to ionize the atom


For the first part the radius is [itex] 5 \times 10^{-11} m[/tex]
the distance between the plates is 0.001 m
divide $5 \times 10^{-11} m/ 0.001 m = 5 \times 10^{-8}$

so far so good, right?

for the second part... since the atom is polarized it has a tiny dipole moment $\vec{p} = \alpha \vec{E}$

$$p = qd = \alpha \frac{V}{d}$$
so $$V = \frac{ q d^2}{\alpha}$$

is that correct??

thank you for your input!

 

[quasar987]

The argument is not all bad imho, but aren't you confusing the use of symbols here? You haven't defined 'd' in the statement of the problem ( ) but I suspect it is just the distance btw the plates. If we call $\vec{\delta}$ instead the distance caracterizing the dipole moment, we have

$$\vec{p}=\alpha\vec{E} \Rightarrow q\delta = \alpha E \Rightarrow V=\frac{qd\delta}{\alpha}$$

and now what? Well we could say that we want V to be such that it induces a dipole moment $q\delta$ such that the work done on the electron in moving a distance $\delta$ parallel to the field is 13.6eV, the ionization energy of the hydrogen atom. So in the end, we need V to be

$$V=\frac{qd(-E_0d/qV)}{\alpha}$$

(E_0=-13.6eV)

This argument has a problem though. It is that the proportionality relation btw the field and the induced dipole holds only for weak fields, such that the deformity is not extreme, as it is in this case since we're literally ionizing the thing.

If I were you I would keep thinking for another way of going at this that does not come with such a defect.

 

[stunner5000pt]

im sorry i didnt define d, sorry

well if we're going to ionize an atom (like hydreogen) what is the minimum value of dipole moment if that exists?

 

[quasar987]

How do you mean "minimal value"?

When there is no field, there is no external, there is zero dipole moment, is that what you mean?

 

[stunner5000pt]

i can see the problem... but i think they are only caring about small pertubations at least with the tools we are given...

 

[quasar987]

ionization is not a small perturbation.

which implies maybe you ought to find a whole other route to the solution.