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!

 

[Nate810]

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.

 

[Moetasim]

I would approach this problem by first understanding the fundamental principles involved. In this case, we are dealing with the interaction between electric charges and the resulting electric field. From the given information, we can assume that the ring is positively charged and the electron is negatively charged. This means that there will be an attractive force between the two charges, described by Coulomb's law: F = kqq/r^2, where k is the Coulomb constant, q is the charge of the ring and electron, and r is the distance between them.

To determine the frequency of oscillation, we need to consider the forces acting on the electron. In this case, we have the attractive force from the ring and the restoring force from the displacement of the electron from the center of the ring. The restoring force can be described by Hooke's law: F = -kx, where k is the spring constant and x is the displacement of the electron from the center.

To determine the spring constant, we need to consider the electric field produced by the ring. The electric field at a point due to a ring of charge can be calculated by integrating the electric field from each infinitesimal element of the ring. This will require the use of calculus, as mentioned in the problem. The resulting electric field equation will be a function of the distance from the ring, r, and the charge of the ring, q.

Once we have the electric field equation, we can calculate the force on the electron at any given distance from the ring. We can then equate this force to the restoring force from Hooke's law and solve for the spring constant, k. Then, using the equation f = 1/(2pi)\sqrt{k/m}, where m is the mass of the electron, we can determine the frequency of oscillation.

Unfortunately, without knowing the specific values for the size and charge of the ring, it is not possible to provide a specific expression or estimate for the frequency. However, by following the approach outlined above, it is possible to determine the frequency of oscillation for any given ring and electron system.

 

[ogb p]

I would first commend your team for taking a systematic approach to understanding the behavior of the electron in this system. It is important to consider the variables and equations involved in order to accurately determine the frequency of oscillation.

Based on the information provided, it appears that the team member's suggestion is to use the equations for electric force (F = kqq/r^2) and simple harmonic motion (F = -kx) to determine the frequency of the electron's oscillation. This approach is valid for small oscillations, where the electron's displacement from the center of the ring is much smaller than the size of the ring.

To begin, we can use the electric force equation to determine the force on the electron due to the positively charged ring. As you mentioned, the distance r from the electron to the ring can be expressed as r = √(x^2 + z^2), where x and z are the coordinates of the electron on the xy and z planes, respectively.

Therefore, the force on the electron can be expressed as F = kQe/r^2 = kQe/(x^2 + z^2). However, this force is not in the form of simple harmonic motion, so we need to find a way to express it in terms of the electron's displacement from the center of the ring, denoted by x.

To do this, we can use the small angle approximation, which states that for small angles, the displacement x is approximately equal to the arc length along the ring. In this case, the arc length is simply r, the distance from the electron to the ring. Therefore, we can rewrite the force equation as F = kQe/r^2 = kQe/(x^2 + r^2).

Now, we can substitute this expression for force into the equation for simple harmonic motion (F = -kx) and solve for the frequency of oscillation, denoted by f.

F = -kx = kQe/(x^2 + r^2)
Therefore, f = 1/(2π)√(k/m) = 1/(2π)√(kQe/m(x^2 + r^2))

This is the expression for the frequency of the electron's oscillation in terms of the size and charge of the ring, as well as the mass of the electron and its displacement from the center of the ring. To estimate the