A charge inside a ring, small oscillation

[ermia]

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

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?

 

[haruspex]

"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##.

 

[ermia]

"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.