Understanding Electron Motion in the Thomson Model of the Atom: Where to Begin?

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The discussion focuses on understanding electron motion within the Thomson model of the atom, specifically how to derive the frequency of an electron's simple harmonic motion inside a uniformly charged sphere. The electric field intensity is given by the formula Qr/4πεR^3 for distances less than the sphere's radius. The force on the electron is expressed as -Qer/(4πεR^3), leading to a differential equation that describes its motion. A participant highlights the importance of including a factor of "r" in the equation to achieve a sinusoidal solution. The conversation emphasizes the need to solve this differential equation to evaluate the frequency of oscillations for the hydrogen atom and compare it with spectral line frequencies.
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Can someone please give me a nudge in the right direction on how to solve this problem? The electric field intensity at a distance r from the center of a uniformly charged sphere of radius R and total charge Q is Qr/4πεR^3, when r < R. Such a sphere corresponds to the Thomson model of the atom. Show that an electron in this sphere executes simple harmonic motion about its center and derive a formula for the frequency of motion. Evaluate the frequency of the electron oscillations for the case of the hydrogen atom and compare it with the frequencies of the spectral lines of hydrogen. How do i even start?
 
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You start with "F= ma". F, the force on an electron with charge e, is -Qer/(4πεR^3) so ma= m (d2r/dt2= -(Qe/(4πεR3)) Can you solve that differential equation? (Note that Qe/(4πεR3[/sup) is a constant.)
 
HallsofIvy said:
You start with "F= ma". F, the force on an electron with charge e, is -Qer/(4πεR^3) so ma= m (d2r/dt2= -(Qe/(4πεR3)) Can you solve that differential equation? (Note that Qe/(4πεR3[/sup) is a constant.)


I think you may have missed a factor of "r" in your differential equation. This is the crucial piece that makes it have a sinusoidal solution.

Zz.
 
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Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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