Question

The capacitance of a infinite circuit formed by the repetition of the same link consisting of two identical capacitors, each with capacitance $C$ (figure), is :

• A. zero
• B. $\frac{\sqrt{5}-1}{2}C$
• C. $\frac{\sqrt{5}+1}{2}C$
• D. infinite

Answer 1

The correct option is B $\frac{\sqrt{5}-1}{2}C$

As seen in the figure, our infinite ladder is composed of infinite identical units. One unit us the part marked with red in this circuit.

As the infinite ladder has a finite, no zero, capacitance, the removal or addition of a number of its identical units does not change its capacitance. This means that our problem can be reduced in the problem of finding the equivalent capacitance of this circuit-unit.

First, let's substitute $C_{eq}$ with x.

x is in parallel with $C_2$, consequently their equivalent capacity is $C_2+x=C+x$.

In turn, $C+x$ is in series with capacitor $C_1$, so the overall equivalent capacity is

$(C+x)C/C+x+C=(1+x)1/(2+x)=(x+1)/(x+2)$

As previously said, the equivalent capacity of this unit is equal to the equivalent capacity of the infinite ladder, which means:

$(x+1)/(x+2)=x\iff x+1=x^2+2x\iff x^2+x-1=0$

$D=b^2-4ac=1+4=5$

$x=(-b\pm D)/2a=(-1\pm \sqrt{5})/2$

As x must be a positive number, $x=(\sqrt{5}-1)/2$

$C_{eq}=(\sqrt{5}-1)/2\mu F$