Find the capacitance of the infinite ladder between points X and Y, figure

Solution

Let, C be the capacitance of the infinite ladder.
As the ladder is infinite, addition of one more elements of two capacitors (1 μF and 2 μF) across the points X and Y should not change the total capacitance. Therefore, total capacity of the arrangement shown in figure must remain C only. The resultant circuit diagram for the circuit is re-drawn as shown in the fig. below.

As seen, capacitor of 2 μF is in series with capacitance C.

$∴$ using the formula, for series combination,
Combined capacity = $\frac{2 \times C}{2 + C}$

This combination is in parallel with 1 μF capacitor.

Therefore,
The equivalent capacity of the arrangement is

   $1 + \frac{2 C}{2 + C} = C$
$\Rightarrow$    $C^{2} + 2 C = 2 + 3 C$
$\Rightarrow$ $C^{2} - C - 2 = 0$
i.e., $C = 2 or, - 1$

Since, the value of capacitance cannot be negative.
$∴$    $C = 2 μF$

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Electrostatic Potential And Capacitance

Find the capacitance of the infinite ladder between points X and Y, figure

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