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

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 $$q_{L2} / q_{R2} = ?$$.

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.

• You mean that these two assumptions cannot be satisfied at the same time, don’t you? I guess the reason is that if wires are too long, then their capacitance cannot be neglected, and so there are some charges on them. Am I right? Okay, lets each wire have some charge, so that they are definitely at the same potentials as corresponding spheres and battery terminals. Then, would I be right if I say that the battery should have constant charge distribution (and the net charge), because only in this case, it has constant potential differences? Commented Jun 9 at 12:49

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.

• Okay, battery can have non-zero net charge, but can I at least say that the battery should have constant charge distribution on its terminal (and then constant net charge), because only in this case, it has constant potential differences? Commented Jun 9 at 12:51
• If you mean the total charge on the battery is the same in the top and bottom pictures, then no, you can't have constant net charge on the battery. The total charge on the battery, plus the charges on the two spheres, plus any charges on the wires will be constant. If you start with zero net charge on the battery, and at the end there's more positive charge on sphere R than negative charge on sphere L, the battery's net charge will become negative. With the spheres far away, the difference in charge on the battery's terminals will be approximately unchanged, but the net charge is not. Commented Jun 9 at 14:19
• Okay, so the net charge of the battery changes during charging the spheres. At the same time, it’s potential difference does not change. For me, it sounds very controversial because I think that new charge distribution on the terminals of the battery should somehow effect on its potential difference … Commented Jun 9 at 14:31
• You can always add charge to the battery without changing the voltage between the two terminals. Imagine a battery-shaped solid conductor, and add charge to it. The two terminals have to be at the same voltage, because it's a conductor. Add that same charge distribution to the battery, and you've added charge without changing the voltage. Commented Jun 9 at 19:35
• Seems like I got the idea. If we add this charge distribution to the battery, potentials on its terminals do change, but potential difference do not. So, we have that potentials themself are not constant, right? I’ve always thought that potential deference ${\phi_{B+}}-{\phi_{B-}}$ of a battery is constant because just ${\phi_{B+}}$ and ${\phi_{B-}}$ are constant. Commented Jun 10 at 6:56

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.

• As I understand, there can be charges of the opposite signs in a conductor only when it is placed into an external electrical field (this process is called charge polarization). Here, I’ve made an assumption that the elements in figure are far from each other, so that we can say each element is isolated, i.e. there is no external electrical field for any element. Where am I wrong? Commented Jun 10 at 5:40
• @Alexandr I agree the isolated spheres can become polarized (that’s in my answer). It’s just that any net charge on each sphere due to the battery would be negligible, since the capacitance would be negligible. Commented Jun 10 at 9:59
• @ Bob D > “I agree the isolated spheres can become polarized”, in my first comment I am saying about exactly the opposite: the isolated spheres can NOT become polarized, cos there is not external field to cause polarization. Is this wrong, or correct? > “… the capacitance would be negligible”, you mean capacitance of the battery terminals? If yes, then how does capacitance of the battery relate with charge on the spheres? I thought charge on the spheres depend only on potentials of the battery terminals and on capacitances of the spheres themselves. Commented Jun 10 at 10:49
• @Alexandr There is an electric field between the positive and negative battery terminals external to the battery. Commented Jun 10 at 11:18
• @ Bob D I don’t know why but sometimes when I ask questions in this site and someone respond me, I get much more new questions, than answers =). Mb, it’s good. Anyway, thanks for your try. Commented Jun 10 at 11:21