Find the capacitance of the infinite ladder of capacitors shown in Fig between points A and B .

Explanation

Let the net capacitance of the given infinite ladder of capacitors between points A and B be C . As the ladder is infinite , addition of one more element of two capacitors `(1 mu F and 2 mu F)` across the points A and B does not change the total capacitance C i.e., value of C should remain unchanged .
Let us do it as shown in Fig.

Here , capacitor of `2 mu F ` is in series with C , hence their combined capacitance will be
`C. = (C xx 2)/(C + 2) = (2C)/(C + 2) mu F `
This combination is in parallel with capacitor of `1 mu F ` , hence , we have `C = C. + 1 = (2C)/( C+2) + 1`
or `C (C +2) = 2C + (C + 2) or C^(2)+ 2 C = 3 C + 2 or C^(2) - C - 2 = 0`
On solving the quadratic equation we find that `C = +2` or `-1 mu F `
However , capacitance cannot be negative . Hence , net capacitance of infinite ladder of capacitors
`C = +2 mu F `