A particle of charge $-e$ orbits a particle of charge $Ze$, what is its orbital frequency?

A point particle $P$ of charge $Ze$ is fixed at the origin in 3-dimensions, while a point particle $E$ of mass $m$ and charge $-e$ moves in the electric field of $P$.

I have the Newtonian equation of motion as $$- \frac{Ze^2}{4 \pi \epsilon_{0}} \frac{\vec{r}}{r^3} = m \ddot{\vec{r}}$$ I derived this from Newton's law and Coulomb's law.

I then went on to show that the particle moves in a plane by showing it has a constant normal vector.

Can anyone help me find this if the orbit is circular about $P$ what it's orbital frequency is -in terms of the constants we have? I would be extremely grateful.

I have attempted this by parameterizing the orbit, differentiating with respect to time and substituting into the equation of motion, however all I get are two unsolvable differentials equations.

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An important point to note are the conserved quantities: you have the angular momentum and the energy conserved (which helps you to reduce the differential equation to a more manageable form). For the circular orbit this is all you need. There is however an additional conserved quantity called the Runge-Lenz vector. – Fabian Oct 9 '12 at 19:11

If the orbit is circular, then $r=R$, $\vec{r} = R(\cos \omega t,\sin \omega t)$, and $\ddot{\vec{r}} = -R \omega^2 (\cos \omega t, \sin \omega t)$. Popping this values into your equation, you get

$$\frac{Ze^2}{4\pi \varepsilon_0 m} = R^3 \omega^2$$.

So your orbital frequency depends on the radius of the orbit, as well as $e$, $Z$ and $m$. This is a variant of Kepler's Third Law of Planetary Motion.

You may want to look at Bohr's Model for a discussion on how this result can be linked with quantun mechanics in the simplest atomic model.

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Thank you! I was trying to solve the parameters as functions of $t$ hence the complicated differential equations, this is very clear now – Freeman Oct 9 '12 at 20:07

This is a standard exercise in Classical physics, Several text books deal with this problem in the context of Gravitational physics. It is same problem with different constants. for example see Landau Classical Mechanics.

The most general orbit is elliptical, Only if there are special initial conditions the particle moves in a circular orbit.

You can easily calculate The velocity required to maintain circular orbit, at a given radius. Apply the concept of centripetal force.

If you have to however solve the problem using quantum mechanics, everything changes.

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