The Energy of a Multiparticle System

[aaj92]

Homework Statement

Consider an electron (charge -e and mass m) in a circular orbit of radius r around a fixed proton (charge+e). Remembering that the inward Coulomb force ( ke$^{2}$/r$^{2}$) is what gives the electron its centripetal acceleration, prove that the electron's KE is equal to -$\frac{1}{2}$ times it's PE; that is, T = -$\frac{1}{2}$U and hence E = $\frac{1}{2}$U. (This result is a consequence of the so called virial theorem. Now consider the following inelastic collision of an electron with a hydrogen atom: Electron number 1 is in a circular orbit of radius r around a fixed proton. Electron 2 approaches from afar with kinetic energy T$_{2}$. When the second electron hits the atom, the first electron is knocked free and the second is captured in a circular orbit of radius r'.

Homework Equations

coulomb force : ke$^{2}$/r$^{2}$

virial theorem T = nU/2

The Attempt at a Solution

I'm not really worried about the second part of this problem quite yet. Right now I'm not really sure how to go about proving that the kinetic energy is -$\frac{1}{2}$ times the potential energy... can someone get me started on the right track?

 

[genericusrnme]

You could use the virial theorem! :p

It's been a while since I've done this but I'm pretty sure it's just a case of writing down the kinetic and potential energies and then stating the correlation.

 

[aaj92]

haha sorry I know it probably seems obvious but we haven't even had the virial theorem mentioned in class. So I'm not really sure how to use it? :/ but I'll try to figure it out

 

[genericusrnme]

Don't worry, the virial theorm is part of action based lagrangian mechanics, you probably haven't encountered it.

You should use the fact that the orbit is circular, that is to say that r doesn't change.
So try playing about with the equation for the inwards force and the outwards centripital (centrifugal? I can't never remember which is the outwards pushing one)
$a_{outwards} = \frac{v^2}{r}$
$a_{outwards} + a_{inwards} = 0$

See if you can work out the kinetic energy from that then see what happens when you compare it to the potential energy

 

[aaj92]

I still can't get this. I'm sorry I'm just slightly frustrated with this and now it's late haha can I just get an explanation for this :/ I'm struggling in this class right now

 

[genericusrnme]

I've been given warnings for being too overzealous with my hinting so I'll try my best to help without giving you the answer


Mod note: removed overzealous answer

 

[aaj92]

Thank you so much! This really helps a lot :)

 

[genericusrnme]

no problem buddy!