Reducing the charge Q on an isolated charged conducting sphere to Q/8

[FranzDiCoccio]

Homework Statement

An isolated conducting sphere carries a charge Q. You want to reduce such charge to Q/8. You can use two further conducting spheres that are identical to the first, but not charged, and some conducting wires for grounding them.

Relevant Equations

- when the spheres touch they share their charge equally, because they're identical
- when one sphere is grounded, the charge on it disperses, and it becomes electrically neutral
- it is possible to induce a charge on a sphere by grounding it, moving it close to a charged sphere, and removing the grounding. The induced charge in this case has opposite sign than the one on the originally charged sphere.

It seems to me that one can obtain the required result by using just one neutral sphere and one ground wire.
Let A be the charged sphere and B be the neutral one. Initially $Q_A=Q$ and $Q_B=0$.

1. put A and B in contact. As a result $Q_A=Q/2$ and $Q_B=Q/2$.
2. ground B, so that $Q_B=0$ again.
3. remove the grounding wire from B and repeat the above steps twice.

It seems to me that this works. After two steps $Q_A=Q/8$.
I have no use for the third sphere and additional wires.
Perhaps the point is that there's a smarter way involving all three spheres, and possibly a smaller number of "operations"?

 

[kuruman]

Yes, you can do it with fewer operations if you use the other sphere. If you had 3 uncharged spheres instead of 2, you could successively touch the sphere of interest to each one without the need for grounding. I think that's the minimum of operations.

 

[FranzDiCoccio]

Hi kuruman,
thanks... I can see what you mean. In that case the grounding wire would be completely useless.
However I have one charged sphere and two uncharged spheres (plus conducting wires), so what you're suggesting cannot be done.
I could use the two uncharged spheres to reduce the initial charge to $Q/4$, but then I'd have to remove the charge from one of the originally uncharged spheres using the grounding wire.
In my opinion this variation is not really something to write home about.