A charge inside a ring, small oscillation

AI Thread Summary
The discussion revolves around calculating the electric field and frequency of small oscillations for a charge inside a ring using Gauss's law. The initial approach involved determining the radial electric field and applying Newton's second law, leading to questions about symmetry and frequency consistency across different axes. There is confusion regarding the relevance of the vertical electric field component and suggestions to focus on potential calculations at small displacements. Participants emphasize the importance of maintaining sufficient terms in approximations to ensure accuracy. The conversation highlights the complexities of the problem, particularly regarding charge density and its implications for the solution.
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Homework Statement
There is an insulator charged ring with linear charge density of ##\lambda =\lambda_0 \sin^2(\theta)##. There is a charge ##q## at the center of the ring. We push the charge forward at x direction ( assuming it is positive ), then we want to find the frequency of small oscillations of the charge. And we do the same thing in y dimension and we want the frequency of small oscillations in this direction too.
Relevant Equations
Gauss law
Laplace equation
Screenshot_20231217_015814_Samsung Notes.jpg

This is the picture of the problem. I attach my solution.
I first used a trick with gauss's law to calculate the radial electric field at first order of r. ( where r is small ) ( we can assume ##small r=\delta r##) I used a cylinder at the center of the ring then i calculated the ##\hat{z}## feild and with that i found the eletric field at r then I used newton second law to find the frequency of small oscillations. Now, the question is why the answer will be the same for every r?! It shouldn't be I think! Because the problem doesn't have symmetry. But my solution gives a radial electric field. Is it true that the y and x frequencies will be equal when calculating to the first order?! Or iam wrong?
 

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"insulator … with … current"?
I confess I do not understand the relevance of the ##\hat z## field. I would solve it by finding the potential at a small displacement d. In making the approximations, you will need to be careful to keep enough terms. I suggest everything up to ##(\frac dr)^2##.
 
haruspex said:
"insulator … with … current"?
I confess I do not understand the relevance of the ##\hat z## field. I would solve it by finding the potential at a small displacement d. In making the approximations, you will need to be careful to keep enough terms. I suggest everything up to ##(\frac dr)^2##.
Sorry I meant charge density.
 
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