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Question
A finite ladder is constructed by connecting several sections of 2 µF, 4 µF capacitor combinations as shown in the figure. It is terminated by a capacitor of capacitance C. What value should be chosen for C, such that the equivalent capacitance of the ladder between the points A and B becomes independent of the number of sections in between?
Solution
The equivalent capacitance of the ladder between points A and B becomes independent of the number of sections in between when the capacitance between A and B is C.
The capacitors C and 4 µF are in series; their equivalent capacitance is given by `C_1 = (C xx 4) / (C+4)`
The capacitors C1 and 2 µF are in parallel; their equivalent capacitance is given by C = C1 + 2 µF
`⇒ C = (C xx 4) / (C+4) + 2`
⇒ 4C + 8 + 2 C = 4C + C2
⇒ C2 − 2C − 8 = 0
⇒ C = −2, C = 4
Capacitance cannot be negative.
∴ C = 4 µF
The value of C is 4 µF.
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