Determining the oscillations of an electron within a sphere

[DaedalusRex]

Homework Statement

Using an electron as a point particle of charge −e inside a positively charged sphere of radius R ≈ 10^(−10) m and total charge +e, find the density ρ(r) of the positive charge for which the electron oscillates harmonically about the center of the sphere assuming that the only interaction involved is electric. Find the angular frequency and its numerical value. Discuss whether this is consistent with the assertion in the previous problem.

Homework Equations

/attempt at a solution[/B]
This is on the first homework set of my EM class, and therefore I have few tools I would be comfortable using to attack this. My idea was to relate the force that the positively-charged sphere (perhaps through Coulomb's Law, F = k((q₁q₂)/r²) ) to the equation of harmonic oscillation, F = -kx. However, I'm uncertain how to move from there. What is the constant in this (if applicable), and how do I determine the angular frequency w = √(k/m)? Also what concerns me is the "density of the positive charge..." I'm completely lost on knowing what that means. Thanks!

 

[kuruman]

Hi Daedalus Rex and welcome to PF.

The density of the positive charge is charge per unit volume, i.e. how many Coulombs per cubic meter you have. To find the frequency, you will need to write Newton's Second Law applicable to this case, then bring the equation to the standard harmonic oscillator form and identify the frequency from that. You need to use Gauss's Law to find the force on the electron at some distance r from the center. Do not use Coulomb's Law.

 

[DaedalusRex]

Hi kuruman, thanks for the quick response.

As I understand it so far, I should be using, as stated in the question, the equation for simple harmonic motion, i.e., x(t) = Acos(wt). What do you mean by using Newton's Second Law for this case, though? Harmonic oscillation through the second law is F = -kx, I'm now thinking I should be relating this equation to Gauss's Law that Flux = (1/E₀)∫ρdv. I feel like I probably have all of the equations I need, but I'm not completely sure how to mesh them all together. Thanks, D

 

[kuruman]

The harmonic oscillator equation is of the standard form $\frac{d^2x}{dt^2}+\omega^2x=0$. What you show as x(t) is the solution to the harmonic oscillator equation. In fact substitute this solution into the equation I gave you and see what you get.

For this problem you need to use Newton's Second Law to find an expression for the second derivative (i.e. the acceleration) then manipulate the equation you get to bring it to the standard form for the harmonic oscillator equation. Note that you need to find the electric field first before you can write an expression for the force on the electron. How is the force on the electron related to the electric field?

 

[rude man]

Equate the centripetal force to the electric field force. The E force is q times the E field. The E field is found from Coulomb's law if you consider the + charge concentrated at the center of the atom.

 

[kuruman]

I don't think that the charge can be considered concentrated at the center of the atom. The electron is "inside a positively charged sphere of radius R". I do not interpret this to mean that it is in orbit outside the sphere. The electric field inside the sphere is not a 1/r² field. The problem as given may be equivalent to the orbiting problem as far as the frequency of oscillations is concerned, but that remains to be shown...

 

[rude man]

I don't think that the charge can be considered concentrated at the center of the atom. The electron is "inside a positively charged sphere of radius R". I do not interpret this to mean that it is in orbit outside the sphere. The electric field inside the sphere is not a 1/r² field. The problem as given may be equivalent to the orbiting problem as far as the frequency of oscillations is concerned, but that remains to be shown...

Agreed. I hadn't noticed the word "inside". Which is in reality absurd since the proton is a very tiny sphere in relation to any electron's orbit. But you're right, in this case the E field must be gotten via Gauss' law.