# Equivalent capacitance of an infinite system

• Fahamedi
In summary, the task is to find the equivalent capacitance of an infinite system between points a and b, where Vi is the potential difference in the number i capacitor. The equations Q=CV and Q_i=CV_i are used to find the relation between Q_i and Q'_i. The potential difference V=Va-Vb is found by looking at the paths in the system, and it is found that V'_i=V_{i+1}+V'_{i+1}. The next step is to divide the circuit into two along the dotted line and find the equivalent capacitance of the righthand portion.
Fahamedi

## Homework Statement

Find the equivalent capacitance of the infinite system between points a and b (see figure).
Where Vi is the potential difference in the number i capacitor.

Q=CV

## The Attempt at a Solution

For number i capacitors we have the relations

$Q_i=CV_i$ and $Q'_i=CV'_i$
$\Rightarrow$ $Q\equiv$ $\sum_{i=0}^{\infty}$$(Q_i+Q'_i)$=C$\sum_{i=0}^{\infty}$$(Vi+V'i)$

Now if V is the potential difference V=Va-Vb and we look at the paths in the system,

$V_0+V'_0=V$ ; $V_0+V_1+V'_1=V$ ; ... ; $V_0+V_1+V_2+...+V_i+V'_i=V$ ; ...

From this I found

$V'_i=V_{i+1}+V'_{i+1}$

$\Rightarrow$ $Q$=C$\sum_{i=0}^{\infty}$$(V_i+V_{i+1}+V'_{i+1})$

And I'm stuck here, I don't know what to do next, or should I do something different?

#### Attachments

• infinity system.png
2.8 KB · Views: 442
Hint: Say you divide the circuit into two along the dotted line. What's the equivalent capacitance of the righthand portion?

## 1. What is equivalent capacitance?

Equivalent capacitance refers to the combined capacitance of a group of capacitors when connected in a circuit. It is a measure of the overall capacity of the circuit to store electrical charge.

## 2. How is equivalent capacitance calculated for an infinite system?

The equivalent capacitance of an infinite system is calculated using a mathematical formula known as the "parallel plate formula." This formula takes into account the distance between the plates, the permittivity of the medium between the plates, and the area of the plates.

## 3. Can the equivalent capacitance of an infinite system be negative?

No, the equivalent capacitance of an infinite system cannot be negative. Capacitance is always a positive value, and in an infinite system, the capacitance can only increase as more capacitors are added.

## 4. How does the distance between the plates affect the equivalent capacitance of an infinite system?

The distance between the plates of an infinite system has a direct effect on the equivalent capacitance. As the distance between the plates increases, the capacitance decreases, and vice versa.

## 5. What is the significance of calculating the equivalent capacitance of an infinite system?

Calculating the equivalent capacitance of an infinite system is important in understanding the behavior and performance of the circuit. It allows engineers to design and optimize circuits for specific purposes and can also help in troubleshooting and problem-solving in complex circuits.

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