You are part of a design team assigned the task of making an electronic oscillat

Question

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. You need to determine how the oscillation frequency of the electron depends on the size and charge of the ring for displacements of the electron from the center of the ring along the axis that are very small compared to the size of the ring. A team member suggests that you first determine the acceleration of the electron along the axis as a function of the size and charge of the ring and then use that expression to determine the oscillation frequency of the electron for small oscillations. Express your answer for the oscillation frequency in terms of the mass (m) and charge (e) of the electron, the charge (q) and radius (r) of the ring, and Coulomb's constant (k). (All letters are lowercase, remember that "e" is a positive constant.) Help Please!!!!

Explanation / Answer

Relevant equations: F = kqq/r^2 F = -kx f = 1/(2pi) k/m----v 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!

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