Electrostatics- spheres leaking charge

[zukkash]

Electrostatics-- spheres leaking charge

Homework Statement

Two small equally charged spheres, each of mass m, are suspended from the same point by silk threads of length d. The distance between the spheres x << d. Find the rate dq / dt with which
the charge leaks off each sphere if their approach velocity varies as v = a / √x, where a is a constant.

Homework Equations

Coulomb's law, F = ma.

The Attempt at a Solution

Approximating the gravity term since x << d, we can write k q^2 / x^2 - mg x / (2d) = m a. We could solve this for q and find dq / dt, etc. However, I only get the correct answer (from the back of the book) if I set a = 0, i.e. the spheres are in equilibrium as they fall toward each other. I also see the assumption of equilibrium made in other solutions of this problem on the web. Why is this assumption justified?

 

[Simon Bridge]

What would it mean for the sphere to not be in equilibrium?

 

[zukkash]

That there's an acceleration, which there should be in general if the velocity follows v = a/√x.

 

[Simon Bridge]

OK - but what does it mean in terms of the charges on the spheres?

 

[zukkash]

Well, that at that time t we have k q^2 / x^2 ≠ mg x / (2d), where q and x both depend on t.

 

[Simon Bridge]

... hmmm... I'm trying to get you to think of the physics.

OK - in the problem, you have taken a bunch of approximations.
You are able to make the approximations ebcause some quantities are very large compared with others.
What has to happen for the approximation for zero acceleration to be valid.
(Note - a zero acceleration would be a constant velocity.)

 

[zukkash]

That the changes in x are small compared to the a in v = a/sqrt(x). Or that the process happens very gradually, so that it's practically in equilibrium throughout.
How would such approximations be justified by the problem statement? It says only that the balls are small and that x << d.

 

[Simon Bridge]

You'd have to look at the broader context of the problem.
Not all the assumptions are stated explicitly.
- i.e. how does the radius of the spheres compare to the geometry? What about the radius of the threads?
The threads are silk - could any of the charge run up the threads - so you'd have to account for the charge distribution there too? Is there, at any time, an uneven charge distribution across the spheres?
Where does this exercise appear in the textbook - in relation to the rest of your education?
Is that "a" supposed to be acceleration: is a a constant with time?
What else is implied?

What does this tell you about the time-scales involved.

 

[zukkash]

Not all the assumptions are stated explicitly.

Yes. But I don't see any assumptions that would justify assuming equilibrium in this case.
I understand reasoning like "the process happens gradually," but I don't see how that is at all justified only on the basis of information given in the problem itself.

- i.e. how does the radius of the spheres compare to the geometry? What about the radius of the threads?

Both of these are small compared to the geometry, which simply allows us to model the spheres as point charges.

The threads are silk - could any of the charge run up the threads - so you'd have to account for the charge distribution there too? Is there, at any time, an uneven charge distribution across the spheres?

How is this justified by the problem statement? This is not a "realistic" problem, it's very artificial; it just involves spheres that are virtually point charges coming toward each other. The mechanism of losing charge is not explained or at all relevant; this is completely artificial and I am wondering if there is any way to explain the assumption of equilibrium in this artificial context, based on the data given in the problem only.

Where does this exercise appear in the textbook - in relation to the rest of your education?

It's a basic exercise in electrostatics. It's exercise 3 in the chapter on electrodynamics in Irodov's "Problems in General Physics." Why is this relevant?

Is that "a" supposed to be acceleration: is a a constant with time?

Oops, I'm sorry, the "a" on the right hand side of the equation of motion I wrote is acceleration, while the "a" in the expression v = a / √x is a given constant. (If a in the expression for v were acceleration, the units certainly wouldn't match up as it is.)

What does this tell you about the time-scales involved.

I just don't see how you could justify assumptions about very long time-scales and equilibrium on the basis of only the information given in the problem itself.

Thanks.

 

[Simon Bridge]

Both of these are small compared to the geometry, which simply allows us to model the spheres as point charges.

In other words, you were able to make a simplification based on an assumption that was not explicitly stated.

How is this justified by the problem statement?

How is what justified in the problem statement?
The point here is to show that there are aspects of the problem that have not been described.
That does not mean they don't matter.

The mechanism of losing charge is not explained or at all relevant; this is completely artificial and I am wondering if there is any way to explain the assumption of equilibrium in this artificial context, based on the data given in the problem only.

That's easy: it is not explainable using only the data provided in the problem statement. You would be expected to use your understanding of the course materials as well.

It's a basic exercise in electrostatics. It's exercise 3 in the chapter on electrodynamics in Irodov's "Problems in General Physics." Why is this relevant?

There is more than one way to skin a cat.
Some methods are ruled out by the available tools.

An exercise in a basic electrodynamics text will have different expectations than the same exercise in a QED text or in a relativistic dynamics text. The author of the problem statement expects you to use the tools already provided in the book, up to that point.

I just don't see how you could justify assumptions about very long time-scales and equilibrium on the basis of only the information given in the problem itself.

You can't.

But you can gain an understanding as to why it may be useful by trying to solve the problem without the assumption.
For instance, without the assumption - is is valid to use electrostatic forces in your model?