Find frequency of oscillating electron in electric field

[gacky27]

Homework Statement

An electron is released from rest in an electric field (see picture attachment, it should explain everything). Upon release, it will oscillate due to the positive electric field. Find the frequency with which it oscillates.

Homework Equations

Coulomb's Law
Equation of an electrostatic field

The Attempt at a Solution

First, I found the electric field. I then used Coulomb's Law to find the force exerted by the field on the electron. I'm a little stuck from this point on.

 

[Redbelly98]

Welcome to Physics Forums.

Please show your results so far! What is the force on the electron that you derived?

Once you have the force, use the assumption z«a,b to get an approximate, simpler expression.

 

[gacky27]

I've actually tried that. It turns out to be zero when I do that, which can't be right. My result for E was:

-(σ/2εo)[(z/b)-(z/a)]

If I plug in z<<a,b I just get zero

My initial train of thought was to find E, use F=qE, then go from there.

Perhaps the value for E i found is wrong? I'll have to go back and redo it I guess.

 

[Redbelly98]

I've actually tried that. It turns out to be zero when I do that, which can't be right. My result for E was:

-(σ/2εo)[(z/b)-(z/a)]

If I plug in z<<a,b I just get zero

My initial train of thought was to find E, use F=qE, then go from there.

Perhaps the value for E i found is wrong? I'll have to go back and redo it I guess.

That looks good so far.

Can you rewrite your expression for E, so that it is

E = something × z​

Eventually, you want to come up with a force for the electron in the form:

F = - k z​

where k is a constant expression involving σ, ε_o, etc.

 

[paranormal]

To find the frequency of oscillation, we can use the equation for simple harmonic motion: f = 1/T, where f is the frequency and T is the period of oscillation. The period of oscillation can be found using the equation T = 2π√(m/k), where m is the mass of the electron and k is the spring constant of the electric field.

To find the spring constant, we can use the equation k = F/x, where F is the force exerted on the electron and x is the displacement of the electron from its equilibrium position.

Once we have the spring constant, we can plug it into the equation for the period of oscillation and then use that to calculate the frequency.

However, it is important to note that the frequency of oscillation may vary depending on the strength of the electric field and the initial position and velocity of the electron. So, the frequency we calculate may not be the exact frequency of oscillation for this specific scenario, but it can give us an estimate.