Gauss' Law & SHM Homework: Proving Simple Harmonic Motion

[Omar Wali]

Homework Statement

According to an old model due to JJ Thompson, an atom consists of a cloud of positive charge within which electrons sit like plums in a pudding. THe electrons are supposed to emit light when they vibrate about their equilibrium positions in this cloud. Assume that in the case of the hydrogen atom the positive cloud is a sphere of radius R = .050nm with a charge of e uniformly distributed over the volume of this sphere. The (point-like) electron is held at the center of this charge distribution by the electrostatic attraction.

a) By deriving its equation of motion, show that when the electron is displaced from its equilibrium position by a distance r, it will execute S.H.M.

Homework Equations

The Attempt at a Solution

∫∫ E ⋅ dA = e(r^3)/(R^3)ε by Gauss' Law (Not sure if enclosed charge is e or 2e)

Solving for E we have ke^2/R^3 * (1/r)

Fe = m(d^2x/dt^2) plug in E and point charge q and we have a second order differential equation that should allow to prove SHM. Unfortunately I'm in high school and we haven't learned how to solve second order differential equations and I'm assuming we're not expected to. I can use a = v dv/dx and solve for v in terms of but I'm not sure how that will help me prove SHM.
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[Omar Wali]

This forum is so helpful! Every time I post a question there's always a response!

 

[TSny]

∫∫ E ⋅ dA = e(r^3)/(R^3)ε by Gauss' Law (Not sure if enclosed charge is e or 2e)

Hi, Omar. You're looking for the E field due to the positively charged cloud. If you think about the total charge of the cloud, that should help you decide between e and 2e.

Solving for E we have ke^2/R^3 * (1/r)

You didn't quite solve for E correctly here. If you can't find your mistake, then please show your steps.

You won't need to solve a differential equation, you just need to argue that the motion will be SHM. You can do this by looking at the mathematical form of the force on the electron in the cloud.

 

[Omar Wali]

Hi, Omar. You're looking for the E field due to the positively charged cloud. If you think about the total charge of the cloud, that should help you decide between e and 2e.
You didn't quite solve for E correctly here. If you can't find your mistake, then please show your steps.

You won't need to solve a differential equation, you just need to argue that the motion will be SHM. You can do this by looking at the mathematical form of the force on the electron in the cloud.

Ok so it will be 2e since we need to add the point charge and total enclosed charge.

And all SHM motions take the form of F= -(constant)r which is what I have minus the negative. But that's a critical component because it's supposed to be a restoring force, no?

 

[Omar Wali]

http://postimg.org/image/qf809shyt/

 

[Omar Wali]

That's all my work so far not sure how to prove SHM.

 

[TSny]

Ok so it will be 2e since we need to add the point charge and total enclosed charge.

No, the E field of the positively charged cloud is due solely to the charge of the cloud. Then, when you put the electron into the cloud, it will feel a force due to the field of the cloud.

And all SHM motions take the form of F= -(constant)r which is what I have minus the negative.

In your first post, you had E = const*(1/r). That would produce a force inversely proportional to r rather than proportional to r.

But that's a critical component because it's supposed to be a restoring force, no?

To see why you get a restoring force, you will need to consider the direction of the E field of the cloud and the sign of the charge of the electron embedded in the cloud.

 

[TSny]

OK, in your hand written notes, your expression for E looks correct.

If you have you studied SHM so that you know that a restoring force proportional to r will cause SHM, then that might be all that you need to state as an answer.

 

[Omar Wali]

No, the E field of the positively charged cloud is due solely to the charge of the cloud. Then, when you put the electron into the cloud, it will feel a force due to the field of the cloud.

Yes sorry this makes more sense now.

In your first post, you had E = const*(1/r). That would produce a force inversely proportional to r rather than proportional to r.

I meant to put it as proportional.

To see why you get a restoring force, you will need to consider the direction of the E field of the cloud and the sign of the charge of the electron embedded in the cloud.

Ok so the gaussian surface at an arbitrary distance r would have E-fields pointing inward which means the electrostatic force works opposite the motion of the electron? Is this a valid statement for why I can make the addition of a negative sign?

 

[TSny]

Ok so the gaussian surface at an arbitrary distance r would have E-fields pointing inward which means the electrostatic force works opposite the motion of the electron? Is this a valid statement for why I can make the addition of a negative sign?

Why do you say that the E field of the positively charged cloud points inward?

 

[Omar Wali]

Why do you say that the E field of the positively charged cloud points inward?

I'm sorry they called the charge of the cloud e which made me think negatively charged. However the e-field is only constant at a certain distance r why would the force restore it to equilibrium?

 

[TSny]

What is the direction of the E field due to the cloud at a point inside the cloud (not at the center)?

 

[Omar Wali]

What is the direction of the E field at a point inside the cloud (not at the center)?

It would be outwards if it is positively charged.

 

[TSny]

Yes. So, what is the direction of the force on an electron inside the cloud?

 

[Omar Wali]

Yes. So, what is the direction of the force on an electron inside the cloud?

Also outwards?

 

[Omar Wali]

Sorry the convention confused me E field lines are in the direction where a positive test charge would move so it will move opposite or inwards

 

[TSny]

Yes, that's right.

 

[Omar Wali]

For the second question I must solve for frequency is it correct to solve for it like this?

$$f = \frac{1}{2\pi }\sqrt{\frac{Constant I found }{m_{e}}}$$

 

[TSny]

When you studied SHM, you probably saw how the frequency is related to the "spring constant" and the mass of the particle.

 

[Omar Wali]

When you studied SHM, you probably saw how the frequency is related to the "spring constant" and the mass of the particle.

Ok I worked out the constant and it has units N/m (which made me smile, finally the end to a problem that's been troubling me for hours!) Thank You!

 

[TSny]

OK. Good work!