# Electron motion on axis of a charged ring

• Ben2
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!

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.

MatinSAR
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.

MatinSAR
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.

MatinSAR

## What is the motion of an electron on the axis of a charged ring?

The motion of an electron on the axis of a charged ring is typically analyzed in terms of the electric field generated by the ring. The electron experiences a force due to this electric field, which can result in oscillatory motion if the electron is displaced from the center of the ring. The specific nature of the motion depends on the initial conditions and the charge distribution of the ring.

## How is the electric field on the axis of a charged ring calculated?

The electric field on the axis of a charged ring can be calculated using Coulomb's law and the principle of superposition. For a ring of radius $$R$$ and total charge $$Q$$, the electric field at a point on the axis at a distance $$x$$ from the center of the ring is given by $$E = \frac{kQx}{(R^2 + x^2)^{3/2}}$$, where $$k$$ is Coulomb's constant.

## What is the potential energy of an electron on the axis of a charged ring?

The potential energy $$U$$ of an electron at a distance $$x$$ from the center of a charged ring is given by the product of the charge of the electron $$-e$$ and the electric potential $$V$$ at that point. The electric potential on the axis of a charged ring is $$V = \frac{kQ}{\sqrt{R^2 + x^2}}$$, so the potential energy is $$U = -\frac{kQe}{\sqrt{R^2 + x^2}}$$.

## Can an electron be in stable equilibrium on the axis of a charged ring?

Yes, an electron can be in stable equilibrium on the axis of a charged ring. If the electron is positioned at the center of the ring (i.e., $$x = 0$$), it will experience no net force because the electric field is zero at that point. Small displacements from the center result in forces that tend to restore the electron to the center, indicating a stable equilibrium.

## How does the charge distribution of the ring affect the electron's motion?

The charge distribution of the ring affects the magnitude and direction of the electric field along the axis, and consequently, the forces acting on the electron. A uniformly charged ring creates a symmetric electric field, leading to predictable oscillatory motion if the electron is displaced. Non-uniform charge distributions can create more complex electric fields, resulting in more complicated electron motion.

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