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
point A and B becomes independent of the
number of sections in between?
(a) 4 μF (b) 2 μF
(c) 18 μF (d) 6 μF

Answer 1

Given:

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.  

To find:

What value should be chosen for C, such that the equivalent capacitance of the ladder between the point A and B becomes independent of the number of sections in between?

Solution:

From given, we have,

The capacitors C and 4 μF are in series.

Cs = 4 × C / 4 + C  = 4C / 4 + C

Thus the capacitors Cs and 2 μF are in parallel.

C = Cs + 2

C = 4C/(4 + C) + 2

C = 4C + 2(4 + C) / (4 + C)

C = (4C + 8 + 2C) / (4 + C)

C = (6C + 8) / (4 + C)

(4 + C) C = 6C + 8

4C + C^2 = 6C + 8

C^2 - 2C - 8 = 0

solving the above quadratic equation, we get,

C = { -(-2) ±√[(-2)² - 4 × 1 × (-8)] } / 2(1)

C = { 2 ± √36}/2

C = (2 ± 6)/2

C = (2 + 6)/2 = 4

Option (a) 4 μF is correct