Periodic motion, electric charge

[tony873004]

Three parallel, infinite, uniformly charged planes are arranged as shown in Figure 24.32.
(it looks just like it's described. The middle plate is positive sigma, the outer plates are both negative sigma.)

Homework Statement

A small hole passes through the middle plane. At t=0 an electron emerges from the hole moving perpendicular to the planes with speed v_o . Assuming v_o is small enough that the electron does not collide with the negative plate, show that its motion is periodic, and find the period.

Homework Equations

The Attempt at a Solution

It's obvious that it will be periodic since there is no force to pull it off the x-axis, and there is no friction to rob it of energy.

I have a feeling it's like a pendulum, where the amplitude does not affect the period. But we haven't had any examples of periodic motion yet in E&M.

From the pendulum, we have a small amplitude formula for period of T=2pi sqrt(L/g). I imagine charge is somewhat analogus to gravity, and perhaps distance between the plates is somewhat analgous to L (length of pendulum). Where do I begin?

Homework Equations

The Attempt at a Solution

 

[rl.bhat]

Using Gauss's theorem calculate the electric field between the plates. Then find force and acceleration of the electron. In SHM, acceleration = w^2.A, where A is the amplitude of SHM and that is the distance between the plates. From w you can find the period of the SHM.

 

[Moetasim]

To begin, we can use the equation for the force on a particle due to an electric field, which is F = qE, where q is the charge of the particle and E is the electric field. In this case, the electric field is uniform and perpendicular to the planes, so the force on the electron will be constant and in the direction of the electric field.

Next, we can use Newton's second law, F = ma, to relate the force on the electron to its acceleration. Since the electron is moving perpendicular to the planes, the acceleration will also be perpendicular to the planes.

Now, we can use the equation for uniform circular motion, a = v^2/r, where v is the velocity of the electron and r is the radius of the circular path. Since the electron is moving in a circular path, we can relate its acceleration to its velocity and radius.

Finally, we can use the equation for the period of uniform circular motion, T = 2πr/v, to find the period of the electron's motion. This equation relates the radius of the circular path, velocity, and period.

Therefore, the period of the electron's motion can be calculated using the equations:

F = qE

F = ma

a = v^2/r

T = 2πr/v

By substituting and rearranging these equations, we can find that the period of the electron's motion is:

T = 2πm/qE

where m is the mass of the electron and q is its charge. This shows that the period is independent of the electron's initial velocity, as long as it is small enough to not collide with the negative plate.