Question

A finite ladder is constructed by connecting several sections of 2 µF, 4 µF capacitor combinations as shown in figure (31-E18). 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 ?

Answer 1

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
$C_1=\frac{C\times 4}{C+4}$

The capacitors C₁ and 2 µF are in parallel; their equivalent capacitance is given by
C = C₁ + 2 µF

$\Rightarrow C=\frac{C\times 4}{C+4}+2$
$\Rightarrow$4C + 8 + 2 C = 4C + C²
$\Rightarrow$C² − 2C − 8 = 0
$\Rightarrow$C = −2, C = 4
Capacitance cannot be negative.
∴ C = 4 µF
The value of C is 4 µF.