#### 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 C_{1} and 2 µF are in parallel; their equivalent capacitance is given by C = C_{1} + 2 µF

`⇒ C = (C xx 4) / (C+4) + 2`

⇒ 4C + 8 + 2 C = 4C + C^{2}

⇒ C^{2} − 2C − 8 = 0

⇒ C = −2, C = 4

Capacitance cannot be negative.

∴ C = 4 µF

The value of C is 4 µF.