- #1

FranzDiCoccio

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- Homework Statement
- A conductive sphere of radius ## R ## carries a free charge ## Q_0 ##. A second conductive sphere is neutral. The spheres are brought into contact. Find the charge on each sphere when they are separated again.

- Relevant Equations
- The formula giving the electric potential outside a charged isolated sphere

Hi,

I think this problem is solved in exactly as a similar problem where the two spheres are very far apart and connected by a very long thin conducting wire. I'm trying to explain this in words, since LaTeX does not seem to work any more (for some reason LaTeX syntax is not replaced by maths in the preview).

So my reasoning is the following.

Also, I wonder whether there is a simpler overall argument. A crucial part of my argument is that the potential on the surface of each sphere is the same when they are infinitely apart. But what is the reason of that, if there is no connecting wire?

Of course the potential on the surface of the spheres (and inside them) is the same when they touch. But in that case (and, in general, when they are not infinitely apart) the charge distribution on their surfaces is far from uniform. Also, I'm not sure the surface potentials stay the same when the sphere are separated.

Thanks for any insight

I think this problem is solved in exactly as a similar problem where the two spheres are very far apart and connected by a very long thin conducting wire. I'm trying to explain this in words, since LaTeX does not seem to work any more (for some reason LaTeX syntax is not replaced by maths in the preview).

So my reasoning is the following.

- In the "experiment" involved in the problem the two spheres actually touch. The charged sphere will loose some of its charge, which will transfer onto the previously neutral sphere.
- While touching, the two spheres form a singular conductor. The charge stays on the surface of this conductor and rearrange so that the electric field inside the conductor vanishes.
- Since all the charge has the same sign, it tends to spread. It won't spread uniformly, though. I expect that there is basically no charge in the vicinity of the contact point.
- When the spheres are separated again, the amount of charge on each of them stays the same. I see no reason for some charge migrating from one sphere to the other.
- The key step is now to assume that the amount of charge on each sphere is the same as in a different "experiment". where the spheres are infinitely apart and are connected through a conducting wire.
- Since there is an "escape route", some charge leaves the originally charged sphere and moves onto the originally neutral one. Another key assumption is that the wire is so thin that the amount of charge that remains on it is negligible.
- Since the spheres are connected, they form a singular conductor, and are therefore at the same potential.
- Since the spheres are infinitely apart, the charge onto their surfaces spreads uniformly. Therefore electric potential on their surface is the same as that of a suitable point charge placed at the center of the sphere. Such potential is proportional to the point charge and inversely proportional to the surface radius.
- Equating the potentials one obtains that
**the ratio of the charges on the spheres is the same as the ratio of their radii**. The final charges are obtained from this condition, along with the fact that their sum should give the original charge.

Also, I wonder whether there is a simpler overall argument. A crucial part of my argument is that the potential on the surface of each sphere is the same when they are infinitely apart. But what is the reason of that, if there is no connecting wire?

Of course the potential on the surface of the spheres (and inside them) is the same when they touch. But in that case (and, in general, when they are not infinitely apart) the charge distribution on their surfaces is far from uniform. Also, I'm not sure the surface potentials stay the same when the sphere are separated.

Thanks for any insight