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Plum pudding model;

-1 electron atom.

-Positive charge of ##+e## distributed evenly about the volume of the atom of radius ##R##.

-Electron (charge ##-e##) is free to move within the sphere.

My first attempt at this is to take it as a 1D problem with a positive line charge of density ##l=\frac{e}{2R}##. When perturbing the electron along the left side of this line of displacement ##-r##, where the origin is at the centre of the sphere, I evaluated the force on the electron from the two sides to be,

$$F_L=-\frac{e^2}{8\pi \epsilon_0 R}\int_{-r}^{-R}r^{-2}dr=-\alpha\left(R^{-1}-r^{-1}\right)$$.

With ##\alpha=\frac{e^2}{8\pi \epsilon_0 R}##, ##F_L## is the force from the left side charge distribution. And for the right side,

$$F_R=-\frac{e^2}{8\pi \epsilon_0 R}\int_{-r}^{R}r^{-2}dr=\alpha\left(R^{-1}+r^{-1}\right)$$.

Where for both I have taken infinitesimal force ##dF##, to be,

$$dF=-\frac{edq}{4\pi \epsilon_0 r^2}$$.

Where ##dq=ldr##. So the net force on the electron (##F_L+F_R##) will be able to give us the equation of motion for the electron, which is

$$\ddot{r}-\frac{\alpha}{m}\frac{1}{r}=0$$.

I'm not sure why but I feel like this is wrong, I would expect the solution to the EOM to be simple harmonic but this doesn't seem to give that. Can anyone clarify whether this is an issue with the 1D approximation, or with the workings? Thanks.