Charge distribution on spheres with varying radii

[jackw]

Homework Statement

Basically, I'm told that two insulated metal spheres, one positively charge (+20uc, sphere A) and one negatively charged (-10 uc, sphere B) come into direct contact (so obviously conduction is the method of charge), and that sphere A's radius is twice the size of sphere B's.

Homework Equations

Not too sure, I know charge usually just reaches an equilibrium when the spheres are identical. A guess would be they both end up at +5 uc but I know that's wrong.

Ra = 2 Rb (lol)

The Attempt at a Solution

Unsure as to how to even start unfortunately. The big question is how the doubled radius on sphere A affects the flow of charge from sphere A to sphere B. This has been bothering me for a few weeks now; help would be appreciated!

 

[haruspex]

Not too sure, I know charge usually just reaches an equilibrium when the spheres are identical.

When what's identical? It's not the charge.

 

[jackw]

When the radii of 2 spheres are identical (ie they have identical symmetrical geometries) but have different charges.

 

[haruspex]

When the radii of 2 spheres are identical (ie they have identical symmetrical geometries) but have different charges.

OK, I see. You meant the charges would become equal if the spheres were identical. They will always come into equilibrium.
But something is equal when the charges are in equilibrium - what?

 

[jackw]

Yeah, and I'm not sure what you mean by the last sentence. I know if the spheres are identical, the charges will become equal when the spheres come into contact. What I'm asking is how do you determine what the new charge on each sphere will be when two non-identical metal spheres (sphere A has a radius twice the size of sphere B's radius; also each sphere has a different charge to start with) come into contact.

 

[haruspex]

Yeah, and I'm not sure what you mean by the last sentence. I know if the spheres are identical, the charges will become equal when the spheres come into contact. What I'm asking is how do you determine what the new charge on each sphere will be when two non-identical metal spheres (sphere A has a radius twice the size of sphere B's radius; also each sphere has a different charge to start with) come into contact.

Yes, I understand that's your question.
Think about this: if the system is not in equilibrium it means the charges will move. What makes them move? They will continue to move until the ... of the two spheres are equal. Fill in the blank.

 

[jackw]

Cool, I would have to guess that they move because the touching objects desire to have charge spread uniformly across their surfaces. I'd also guess they continue to move until the charges of the two spheres are equal? So you're saying regardless of varying radii sizes, the spheres will reach the same charge? But if this is true, how would one quantitatively work out what this new charge will be (given the original two charges)?

 

[haruspex]

Cool, I would have to guess that they move because the touching objects desire to have charge spread uniformly across their surfaces.

No, as I wrote previously, it is not the charge that must equalise.
A charged particle moves in response to an electric field. An electric field results when there is a potential difference. If the charges are in equilibrium then there's no tendency to move, so no field, so no ... what?

 

[jackw]

So there's no potential difference when the charges are in equilibrium? But what does this have to do with the question? I have no idea how "potential difference = 0" could affect the final charges of spheres with varying radii.

 

[haruspex]

So there's no potential difference when the charges are in equilibrium? But what does this have to do with the question? I have no idea how "potential difference = 0" could affect the final charges of spheres with varying radii.

It means the two spheres must be at the same potential.
That said, I do not know how to use that to find the charge distribution between the spheres. One could crudely use the formula for the potential of each sphere taken in isolation, but that fails to take into account the contribution each makes to the potential of the other.
The paper at http://www.fmf.uni-lj.si/~podgornik/download/Lekner-attraction.pdf refers to an 1891 paper by Maxwell, and quotes quite a complicated formula. See page 8 of the PDF.

 

[jackw]

It seems odd that the problem would require advanced knowledge/formulas as this came from a high school physics textbook. Also, I didn't state specifically in my initial post, but the question asks for the new charges on each sphere after they're touched. It also gives a tip (charge resides equally distributed over the surface of a spherical metallic conductor).

 

[haruspex]

It seems odd that the problem would require advanced knowledge/formulas as this came from a high school physics textbook. Also, I didn't state specifically in my initial post, but the question asks for the new charges on each sphere after they're touched. It also gives a tip (charge resides equally distributed over the surface of a spherical metallic conductor).

That tip is correct when there are no other charges present. Two spheres near each other don't satisfy that, let alone two spheres in contact.
As I said, you could, crudely, pretend that is the situation. Suppose the charges the spheres get are Qa, Qb. You know Qa+Qb. If a conducting sphere radius Ra has charge Qa, no other charges present, what is the potential at that sphere?