Electron motion on axis of a charged ring

[Ben2]

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!

 

[berkeman]

You should fix your LaTeX and upload a diagram of the problem...

 

[haruspex]

Is this what you meant?

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!

 

[kuruman]

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?

 

[Ben2]

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!

 

[berkeman]

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.

 

[kuruman]

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.

 

[haruspex]

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.