Electron motion on axis of a charged ring

AI Thread Summary
The discussion revolves around demonstrating that an electron can oscillate harmonically along the axis of a charged ring, with the relevant frequency formula provided. The equation for the electric field is noted, but there seems to be confusion regarding the application of harmonic motion principles and the derivation of the frequency. Participants emphasize the need to clarify the connection between the force on the electron and its displacement to establish simple harmonic motion. Additionally, there are suggestions to improve LaTeX formatting and upload diagrams for better understanding. The conversation highlights the importance of correctly applying equations and concepts from physics to solve the problem effectively.
Ben2
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Homework Statement
"An electron is constrained to move along the axis of the ring of charge in Example 5. Show that the electron can perform oscillations whose frequency is given by ##\omega = \sqrt{frac{eq}{4\pi(\epsilon_0)ma^3}}##. This formula holds only for small oscillations, that is, for x<<a in Fig. 27-10. (Hint: Show that the motion is simple harmonic and use Eq. 15-11." [Halliday & Resnick, Ch. 27, Problem 19]
Relevant Equations
##K=frac{1}{2}(kA^2)\sin^2((\omega)t+\delta)## (15-11)
Showing the motion is simple harmonic seems routine. The 5th equation on p. 674 gives ##E=frac{1}{4\pi\epsilon_0}frac{qx}{(a^2)+(x^2)}^frac{3}{2}##, but matching expressions for ##\omega=k/m## yields only ##x=frac{ea^2}{2}##. Something in the model is escaping me. Thanks for any help offered!
 
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You should fix your LaTeX and upload a diagram of the problem...
 
Is this what you meant?
Ben2 said:
Homework Statement: "An electron is constrained to move along the axis of the ring of charge in Example 5. Show that the electron can perform oscillations whose frequency is given by ##\omega = \sqrt{\frac{eq}{4\pi(\epsilon_0)ma^3}}##. This formula holds only for small oscillations, that is, for x<<a in Fig. 27-10. (Hint: Show that the motion is simple harmonic and use Eq. 15-11." [Halliday & Resnick, Ch. 27, Problem 19]
Relevant Equations: ##K=\frac{1}{2}(kA^2)\sin^2(\omega t+\delta)## (15-11)
Ben2 said:
Showing the motion is simple harmonic seems routine. The 5th equation on p. 674 gives ##E=\frac{1}{4\pi\epsilon_0}\frac{qx}{(a^2+x^2)^\frac{3}{2}}##, but matching expressions for ##\omega=k/m## yields only ##x=\frac{ea^2}{2}##. Something in the model is escaping me. Thanks for any help offered!
 
Ben2 said:
Showing the motion is simple harmonic seems routine.
Equation 15.11 is probably derived in relation to a spring-mass system (chapter 15) and is related to harmonic motion. This problem is in chapter 27. I am not sure what the "routine" procedure you followed to show that the motion is harmonic because you don't show it. Perhaps you should show it.

Frankly, I don't see the usefulness of the hint. Once one shows that the motion is simple harmonic, what additional ask does the statement "and use Equation 15.11" require one to do? Am I missing something?
 
Thanks for the comments. For Mentor berkeman: Pending a computer cleanup, I've held off on installing a LaTeX translator (Is MathJax what you want?). On uploading diagrams, I'm a complete dunce. For haruspex: That is all correct. For kuruman: Besides the form of H&R's second equation p. 674, ##\cos\theta = frac{x}{\sqrt{a^2+x^2}}## (apologies for more fractured LaTeX), I don't see a connection with simple harmonic motion. The diagram is p. 673 Fig. 27-10, also found in Problem 569 of "The Physics Problem Solver." Thanks again!
 
Ben2 said:
For Mentor berkeman: Pending a computer cleanup, I've held off on installing a LaTeX translator (Is MathJax what you want?).
You don't need to upload anything -- PF takes care of all of the rendering for you.

I'll send you a DM with tips on using LaTeX here.
 
Ben2 said:
Thanks for the comments. For Mentor berkeman: Pending a computer cleanup, I've held off on installing a LaTeX translator (Is MathJax what you want?). On uploading diagrams, I'm a complete dunce. For haruspex: That is all correct. For kuruman: Besides the form of H&R's second equation p. 674, ##\cos\theta = frac{x}{\sqrt{a^2+x^2}}## (apologies for more fractured LaTeX), I don't see a connection with simple harmonic motion. The diagram is p. 673 Fig. 27-10, also found in Problem 569 of "The Physics Problem Solver." Thanks again!
The connection you want to establish is this. Show that the force on the electron for small displacements ##z## from the origin on the z-axis is restoring and proportional to the displacement ##z##. The constant of proportionality is the frequency squared.
 
Ben2 said:
apologies for more fractured LaTeX
Mostly it is that you are omitting the \ in front of frac.
In post #1, you also had the 3/2 exponent in the wrong place.
 
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