How do charges on two conductors charged by the same battery relate to each other?

Question

Let’s consider 2 neutral conducting spheres, L and R, with different radii. There is a battery somewhere in the middle between these spheres. The battery has potential $\phi_{B-}$ on its negative terminal and $\phi_{B+}$ on its positive terminal.

Let’s make the following assumptions:

1. All the elements are located at quite large distances from each other, so that the electric fields of these elements don’t affect the charge distribution on these elements.
2. Two wires, which we’ll use later to connect the battery with spheres, have a low capacitance, so that we can neglect charges on these wires.

Now, using the wires, let’s connect sphere L with the negative terminal of the battery and sphere R with the positive terminal. The process of charging is shown in the figure below.

There is a contradiction I can’t resolve.

1. When the conductors are fully charged, each must be at the same potential as the corresponding terminal of the battery, $\phi_{B-}$ and $\phi_{B+}$. The charges on the spheres in generally should be different in absolute value, i.e. $|q_{L2}| \neq |q_{R2}|$ because, the potentials on the spheres in generally are different in absolute value, $|\phi_{L2}| \neq |\phi_{R2}|$ or $|\phi_{B-}| \neq |\phi_{B-}|$, and the capacitances of the spheres are different $C_L \neq C_R$.

2. We suppose that the battery sustains constant potential difference $\phi_{B+} - \phi_{B-}$. Then, if I understand correctly, there should be a constant charge distribution on the battery terminals. Therefore, using charge conservation law, we should conclude that at any moment the charge left the negative terminal of the battery should be exactly equal (with minus sign) to the charge arrived at the positive terminal. Finally, here we can conclude that at any time moment (particularly, when fully charged), spheres should have opposite charges $q_{L2} = - q_{R2}$.

So, these two reasonings cannot be correct at the same time. I suspect the problem is in the second one, but why? I also would like to know how charges on the sphere relate to each other in this case $q_{L2} / q_{R2} = ?$.

Answer 1

Not all of the assumptions that you have imposed can be satisfied.

Let's look at All the elements are located at quite large distances from each other and $\dots$ we can neglect charges on these wires.

Looking at you last diagram you are implying that bringing a test charge from infinity to the right hand side of the system of two spheres, a battery and two connecting wires is independent of path taken by the test charge and will take the same amount of external work because all those parts are at the same potential.

Something not quite right here as your assumptions imply that less external work needs to be done in bringing the test charge to the middle of the connecting wire than to either the positive terminal of the battery or the right hand sphere.

Answer 2

Assumption 2 is incorrect. You can't say you'll assume thin wires, and so neglect the charge. However thin the wires are, the electric field along the wires must be zero, which is what leads to each sphere being at the same potential as the corresponding battery end. With no charge on the wires, the field along the wires can be non-zero.

Beyond that problem, your reasoning that "using charge conservation law, we should conclude that at any moment the charge left the negative terminal of the battery should be exactly equal (with minus sign) to the charge arrived at the positive terminal." is incorrect. The total system charge is conserved, but the battery itself can have an overall non-zero charge.

Answer 3

I have a bit of a different take on this.

The degree to which the two spheres will become "charged" will depend on the degree to which the spheres are physically isolated from one another.

If they are isolated such that their separation is much greater than their radii, the primary effect of the battery will be to redistribute the existing charge, causing electrons on sphere R to move toward the side of the sphere connected to the positive battery terminal, and repelling electrons on sphere L causing them to move aways from the side connected to the negative battery terminal. See the upper figure below. The net charge on each sphere essentially remains zero.

On the other hand, if they are in close proximity as shown in the bottom figure, such that the distance between them is much less than their radii, they will constitute a form of a capacitor (in this case an air gap capacitor). Then the positive terminal of the battery will remove negative charge from Sphere R giving it a net positive charge, while the negative terminal of the battery will deposit an equal amount amount of negative charge on Sphere L giving it a net negative charge. This results in an electric field (albeit non uniform) between the spheres that stores electrical potential energy.

Hope this helps.