Why does charge distribute evenly between two halves of a conducting sphere? Always while studying I was told that if two identical spheres, one charged and the other uncharged, are brought together the charge in each one of them is halved.
However, why does it happen?
Is it because the electrons repel each other in the the charged sphere and shift to the other sphere, which is uncharged?
For example lets consider two spheres of radius $12\text{ cm}$ and $6\text{ cm}$, each having a charge of $3\times10^8\text{ C}$. Then why would the current flow from the sphere of $6 \text{ cm}$ to the sphere with $12\text{ cm}$ of radius?  
Till what time will a current flow? i.e till the charge density is equal, the electric field is equal or the the force of repulsion is equal.
 A: The basic condition that has to hold inside a conductor is that the electric field vanishes everywhere, i.e. $\vec{E}=0$. If you join the two halves of the sphere, the electrons redistribute themselves in such a way that the configuration reaches equilibrium. The latter state can only be reached with vanishing electric field, otherwise there would be a force between the electrons causing acceleration.  
As a consequence, the electrons have to distribute evenly, which results in half of the charge being carried by each half of the sphere. 
Vanishing electric field implies constant potential $V$, as seen from the following picture: 

A: Let's consider each sphere have positive charge of amount $3\times10^8\text{ C}$. Obviously, the smaller sphere has more charge density. Now when the two sphere are being brought close to each other,obviously electric field will be from lower sphere surface to larger sphere surface. So the potential will be large at the small sphere point with respect to the large sphere point where you going to meet the two sphere ($\vec{E}=-\vec{\nabla}\phi$) . (remember,as a side note, during the motion, the potential of the any individual conductor will be not be same at every point on it  and hence charges move. If you stop moving them, any conductor will be again be equipotential, when the charges redistribute themselves in such way that there is no potential difference in the conductor.)  
Now, when you meet them with each other, obviously positive charges (actually electrons will move from larger sphere to smaller sphere, to have less potential energy ) on the smaller sphere will go to the larger sphere to have lower potential energy, and this  will go on until the potential at every point on the two sphere becomes equal. Obviously the charge distribution will no longer be uniform as the electric field in the space will have a complicated nature( try to visiualize your case, from this) and you know $$E_{just\_out}=\frac{\sigma}{\epsilon_0}$$
