Charge Flow Between Two Spheres Different in Sizes

Question

I'm struggling to understand the following concept:

Given two conducting metal spheres of different radii, each have charge -Q (the same charge). If the two spheres are to be connected by a metal wire, charges will flow from the smaller sphere to the bigger one.

From what I understand, one of the requirements for charges to flow is that there must be a potential different between two points. So this is my calculation for the potential different between these two spheres: $$\Delta V = Ed$$ $$\Delta V = 0 \cdot d=0$$ I reasoned that the field between these spheres is zero because the vectors are opposite, therefore they "cancel" each other out. But then how can the charge flows at all, given that there is no potential difference between these two spheres?

(Sorry if I miss a tag. I'm new to Physics and don't know most of the topics).

Answer 1

Let me suggest an analogy. Suppose we take two spherical containers, one with a volume of one litre and the other with a volume of one cubic metre, and put a mole of gas in each. Then we connect them with a pipe, and we find that gas flows from the smaller to the larger container.

The reason for this is obviously that if you squeeze a mole of gas into a 1 litre container you get a higher pressure, while the same mole of gas in the larger container has a lower pressure. Then when you connect the spheres gas flows from the high to the low pressure.

The point of this is that it applies to squeezing charges onto spheres in the same way. When you squeeze a charge $Q$ onto a sphere of radius $R$ it raises the potential of the sphere - the potential is equivalent to the pressure in our analogy. Specifically the potential of the sphere is given by:

$$ V = \frac{Q}{C} $$

where $C$ is the self capacitance of the sphere:

$$ C = 4\pi\epsilon_0 R $$

So when you "squeeze" the same charge $Q$ onto a two spheres with different radii the smaller sphere has a large potential. Hence when you connect the spheres by a wire charge flows from the smaller sphere to the larger.