What Is the Frequency of Electron Oscillations Near a Charged Square?

[anap40]

Homework Statement

A square of side a located in the x-y plane and centered on the origin carries a total
charge Q uniformly distributed over its circumference.
(a) What is the electric eld at any point on the z-axis? How does the eld behave far from
the square?
(b) An electron constrained to move along the z-axis near the center of the square is seen to
exhibit small oscillations above and below the plane of the square. What is the frequency of
these oscillations? (assume Q is positive)

Homework Equations

So for the first part I got E=KQz/[(x^2+y^2)^2sqrt((x^2+y^2)^2+(a/2)^2)]

For the behavior far from the square I made z>>a/2 and got back E=kQ/b^2

I am stuck on part B. The force is just q times the field, and z<<a/2 but I don't know where to go to get the period.

I appreciate any help.

 

[belliott4488]

You should reconsider your answer for (a). If someone gives you a location on the z axis, you should be able to return with the value of the field at that location. Your expression for E depends on x and y, however - what are they? You can't provide an answer since you have no values defined for them. In short, your answer must depend only on given constants (like a and Q) and the dependent variable(s), in this case, z.

What you need to do is to sum the contributions of all the charges along the sides of the square.

 

[anap40]

OK, thanks for pointing that out. I re did it and I got what I think is probably the correct answer.(it wasn't that much different than the other one, I just change one varable into terms of a before integrating.)

E=KQZ/[(z^2+(a/2)^2)(z^2+(a^2)/2)^.5]

So, now when making z>>a I get the formula for an elctric field of a particle.

I think that to get the period of the oscillation I need to find out what happens when z<<a, but all i get is that it goes to zero.

Immediatly around zero, a graph of the field shows that it is approximatley a straight line. I know there is some way get an estimate of what that line is using calculus, but I don't remember how. Is it with a taylor series expansion?

Thanks for the help.

 

[belliott4488]

Can you say how you got that expression for the E field? It's not what I get.

However it works out, though, you always get the period of small oscillations in the same way. Do you remember working on the problem of a mass on a spring, i.e. the simple harmonic oscillator (SHO)? That's important because 1) you can solve it exactly and 2) if you expand the expression for any restorative force (i.e. one that has an equilibrium point) in a Taylor series about the equilibrium point, then for small the displacements only the leading term is significant, and it is linear, i.e. it looks just like the SHO. You'll see this over and over in Physics - any weird restoring force will look like a SHO for small oscillations. (You already did this for a pendulum, I'm sure.)

 

[anap40]

Here is my work.
The last step was to sub back in for b^2 and multiply by 4 (because there are 4 sides of the square)

http://img256.imageshack.us/img256/1089/imgxj3.jpg

 

[belliott4488]

Sorry - I got called away and haven't been able to look at this until now. I think your solution for the E field looks right - I misread it the first time.

I'd try expanding that in a Taylor series about z=0 next, and then look at the leading terms for small values of z.