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The effective capacitance between A and B of an infinite chain of capacitors joined as shown in the figure - <img class=img-fluid question-image alt=Question src=https://cdn.tardigrade.in/q/nta/p-3rsyw7scm39id6pk.jpg />
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Q. The effective capacitance between $A$ and $B$ of an infinite chain of capacitors joined as shown in the figure -
NTA Abhyas
NTA Abhyas 2022
A
$\left(\sqrt{3} - 1\right)C$
B
$\frac{\left(\sqrt{5} - 1\right)}{2}C$
C
$\left(\sqrt{5} - \, 1\right)C$
D
$C$
Solution:
$C_{e q}=\frac{C \left(\right. C + C_{e q} \left.\right)}{C + C + C_{e q}}$
$= \, C^{2} \, + \, CC_{e q}= \, 2CC_{e q} \, + \, C_{e q}^{2}$
$C_{e q}^{2}+CC_{e q}-C^{2} \, = \, 0 \, \Rightarrow C_{e q} \, = \, \frac{- C \pm \sqrt{C^{2} + 4 C^{2}}}{2}$
$= \, \left(\frac{\sqrt{5} - 1}{2}\right)C$
