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Equivalent capacitance of an infinite system

  1. Jun 21, 2011 #1
    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

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

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

    [itex] V_0+V'_0=V[/itex] ; [itex]V_0+V_1+V'_1=V[/itex] ; ... ; [itex]V_0+V_1+V_2+...+V_i+V'_i=V[/itex] ; ...

    From this I found

    [itex] V'_i=V_{i+1}+V'_{i+1}[/itex]

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

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

    Attached Files:

  2. jcsd
  3. Jun 21, 2011 #2

    vela

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