P2) A uniformly charged sphere of radius R has an electricfield
3
E r
R
= for r < R.
Consider a very simplified model of the atom where the positivenucleus, consisting of Z
protons, is a point charge at the center of the atom, surrounded bya uniform cloud of Z
electrons.
(a) If the nucleus is displaced relative to the electron cloud by asmall distance x, what is
the force acting on the nucleus due to the cloud (and equivalently,the force acting on the
cloud due to the nucleus)?
(b) In PHYS 132 you will have seen simple harmonic oscillators,such as a mass on a
spring. It can be shown that if the acceleration a of an object isdirectly proportional to
its displacement x from a point of equilibrium, then the objectwill undergo simple
harmonic motion with a frequency:
2
f ?
p
= , where 2
a x
?
=-
Determine the frequency of oscillation if the nucleus is displacedrelative to the center of
the electron cloud and then let go. Hint: from knowing the massesof the protons and
electrons, is it the cloud that primarily moves back and forth oris it the nucleus?
Calculate this frequency for a hydrogen atom; you can find theatomic radius R on
Wikipedia. If this oscillating atom emits an electromagnetic waveat the same frequency,
what is its wavelength?