Charged ring and electric field problem

[Ertosthnes]

The problem:
You are part of a design team assigned the task of making an electronic oscillator that will be the timing mechanism of a micro-machine. You start by trying to understand a simple model, which is an electron moving along an axis through the center and perpendicular to the plane of a thin positively charged ring. A team member suggested to determine how the frequency of the electron depends on the size and charge of the ring for displacements of the electron from the center of the ring which are small compared to the size of the ring. Follow through with this suggestion and determine if an expression for the oscillation frequency of the electron for small oscillations can be determined by such an approach. If so, provide the expression, and, for reasonable values for the size of the ring and its charge, estimate the frequency.

Relevant equations:
F = kqq/r^2
F = -kx
f = 1/(2pi)$$\sqrt{k/m}$$

Calculus is needed to determine the sum of the electric field.

For variables, I'm setting the ring on the xy plane with the electron above it on the z plane. The distance from the electron (charge e) to the ring (radius R) at any point is r = $$\sqrt{x^{2}+z^{2}$$.

For force, so far I have F = KzQe/R^3, but I'm not even sure that's right. Somehow, I need to put frequency in terms of R and e. Any help would be much appreciated!

 

[member 553083]

Solution:The electric field of a thin ring of charge is given by the equation:E = \frac{Kq}{R^2}where K is Coulomb's constant, q is the charge of the ring, and R is the radius of the ring.The force on an electron of charge e in the presence of this electric field is given by the equation:F = K e q/R^2This force can be written in terms of the displacement of the electron from the center of the ring (x) as:F = -K e x/R^3We can use this equation to calculate the frequency of oscillation of the electron around the ring. Since the equation of motion for the electron is given by F = ma, we can use this to determine the angular frequency of oscillation (f) of the electron about the ring:f = \sqrt{\frac{K e}{m R^3}}Where m is the mass of the electron. For reasonable values of the size of the ring (R) and the charge (q), the frequency of oscillation can be estimated. For example, if the radius of the ring is 1 cm and the charge is 10 nC, the frequency of oscillation would be approximately 10^14 Hz.