The Infinite Array of Electrical Capacitors

We are talking about an association of so many capacitors or capacitors of identical capacitance in such a way that hypothetically their junctions form an infinite mesh.

Initially, the array is uncharged. Then a power battery is connected to one of the pairs of junctions, say A-B.

The question is to know what will be the value of the electric potential at any of the mesh junctions?

Curious isn’t it?

Let’s start from the premise that since the array is infinite, then each generic element i will contain a quantity of charge qᵢ.

In this way we can apply the definition of capacitance to a junction of the array let’s assume the junction P and its four branches.

The theory of electricity defines capacitance as:

i.e,

Applying this to each element and with the support of the drawing above, we have:

Also, by the theory of conservation of charge, if we consider a closed region around the point P involving each charge of the four branches (restricted to only one plate of each capacitor) the total sum of them will be equal to zero, since it is assumed the system in equilibrium. Then, we have:

and

that simplifying

or

and finally

✅We can then even write as a theorem that:

The potential at any junction of an infinite array of identical capacitors is equal to the arithmetic mean of the potentials of the four closest junctions.

📌 Note that we do not mention signs of electrical charges, as their values ​​are duly treated as independent variables.