Electrostatic oscillator

[Jacques]

Hi

I have a question about some theorical setup and try to understand how it will behave. The goal of that exercise is to help me acquire some knowledge of how to apply the electromagnetism math to concrete setup.

All setup are in empty space (no air) where gravity is negligible, in a farady cage and magnetically sheilded.

First set up made up with 2 non-conducting objects.

- A point like sphere S of charge Q1 and mass m1

- A perfect ring O of charge Q2 and radius R and mass m2

The center of O is at the origin and lay on xy plane

S is at a distance z on the z axis

What will happen if we keep O stationary and let S go.

It depend of the charge of Q1 and Q2.

If they are the same S will be repelled by O.

If they are opposed S will be attracted to O. The geometry of O will give an acceleration to S toward the O center. The potential energy will be transformed in kinetic energy until S pass the origin. After that S will decelerate and the kinetic energy will transformed in potential energy until the kinetic energy is 0. S will accelerate back to the origin etc etc.

So S will oscillate on the z axis.

What is the frequency of that system ???

Also accelerated charge produce radiation and the system will loose energy.

What is the damping of this oscillator ???

 

Setup two: same as setup one but with O made of a conductor. That's where it will get complicated because the motion of S create a magnetic field that will induce an electric current in O that will create a magnetic field that will modify S motion... But let concetrate on the first problem.

 

How:confused: do I start ?

[swansont]

You should write down the equation for the force on the particle, S.

 

F = ma = m d^2x/dt^2

 

Then you solve the differential equation for the motion of S.

 

What is the force?

[Jacques]

Hi Swansont

Thanks for the answer but I figure out that I don't have time to figure out all the maths (family, work, girl friend...) but I will try to give what I know.

The force on S is the summation of all forces between S and all elements of O. That will give the total force in the z direction since the force in xy plane all cancel out. That force will make S move toward the origin and the force will increase (because of the inverse square law) so need to intergrate over t also. that about it . I can see that a charge particle will oscillate.

That bring me to an other question : If the particle have x and/or y different from zero but smaller than the ring, will the particle oscillate or will it hit the ring ?

Also, how can we charge a non conductor ?

Thanks

[swansont]

If you are off-axis you will have a force toward the closest part of the ring.

 

Non-conductors can be charged (ever charge up a balloon by rubbing it on your head?) but the charge will not generally distribute itself over the surface to make the electrostatic potential equal everywhere.

[Jacques]

If you are off-axis you will have a force toward the closest part of the ring.

OK Does that mean that in linear accelerator you need magnet to keep the beam focused ? Just electrostatic wouldn't be enaught to keep the beam together.

For the charging, that what I was expecting.

 

Do you know if there are free electrostatic simulation program ?

Thanks

[swansont]

In an accelerator you'd have more than a ring, though — you can have the whole tube at some potential, and so that it's repulsive and would tend to confine the beam. That way you would get oscillations about the axis, for anything that was off-axis.

[Jacques]

The field would look like that I guess. I found a free program to trace electric field Electric Field It is 2D but it is free. I would like to find something similar in 3D

I putted some charge but I am not sure if they are the best...

That way you would get oscillations about the axis, for anything that was off-axis.

Will there be an appreciable damping of the oscillation caused by EM radiotion of the charged particle? What are the factor causing the damping ?

Thanks