Question

. Assume that we live in a nineteenth-century world where no one ever heard anything about
quantum mechanics. Classical laws work perfectly well, even at the sub-atomic scale. An
experimental physicist tries to build up a hydrogen atom by placing an electron in a fairly
circular orbit around a proton. The total power radiated by this non-relativistic electron as it
accelerates around the proton is given by Larmor’s formula, in SI units,
P =
dE
dt
= −
1
4πε0
2q
2a
2
3c
3
, (Q1-1)
where q is the electronic charge, a is the acceleration of the point charge (in this case, an
electron), and c is the speed of light.
(a) (4 points) Eq. (Q1-1) shows energy lost per time during each revolution around the proton.
Assume that acceleration is centripetal, find the change in energy per orbit.
(b) (4 points) Using the result of Part (a), show that energy lost per orbit is negligible com-
pared to the kinetic energy of the electron.
aaa
(c) (a points) Radiation electron will make spiral orbits around the proton. However, does
the answer of Part (b) justify Eq. (Q1-1) to give an excellent approximation that orbit of
the electron remains fairly circular at any instant?
(d) (4 points) Derive a relation that how much time will the electron take to spiral around
the proton before falling into it?
(e) (4 points) Using the formula derived in Part (d), what would be the lifetime of the electron
if the size of the atom is 1 Å and of the nucleus is 1 fm.
(f) (4 points) What would be the revolving electron’s velocity for the atom given in Part (e)?
Compare your results with the speed of light and write your inference. Does it justify the
assumptions made in Part (a)?
(g) (4 points) Before falling into the proton, what is the minimum amount of energy this
revolving electron can have? Would there be any lower limit to this energy? If no, then
why not?

Answer 1

Answer:

that is a tough question