How Does a Charged Ring Affect Electric Field and Oscillation Frequency?

[serrakartal]

Homework Statement

A uniform circular ring of charge Q= 6.40 microCoulombs and radius R= 1.30 cm is located in the x-y plane, centered on the origin as shown in the figure.

Homework Equations

1.What is the maximum value of E on the z-axis?

2.What is the frequency of the small axial oscillations that the electron will undergo if it is released along the z-axis near the origin?

thanks for helping already

 

[24hourtutor]

Maximum Value of E:

1. Determine the direction of the E-field on the z-axis. This should be clear from the symmetry.
2. Look at a small "dq" of charge on the ring. At some position "z," above the ring, what is this dq's contribution "dE" to the total electric field? Remember what you did in step 1. Only a component of dq's E-field actually goes towards the total E-field that we care about. Use some trig!
3. Once you have dq's contribution, it is time to add up ALL the dq's (it's integral time).
4. Since you solved for the E-field at some arbitrary "z," you now know the value of E for ANY z! Now you just have to find the "best" z.

Frequence of Small Oscillations:

Whenever you see a question like this, you should think "what is the potential energy in this neighborhood." I'm not going to write it all out, but some tools to consider using: are a Taylor expansion around z = 0 and then trying to get the potential energy to somehow look like the PE of a spring. If you can do that, you've basically solved the problem.

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