# Equivalent capacitance of an infinite system

1. Jun 21, 2011

### Fahamedi

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
Q=CV

3. 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?

#### Attached Files:

• ###### infinity system.png
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2. Jun 21, 2011

### vela

Staff Emeritus
Hint: Say you divide the circuit into two along the dotted line. What's the equivalent capacitance of the righthand portion?