Chapter 4: Q83CP (page 879)

26.83 An Infinite Network. As shown in Fig. P26.83, a network of resistors of resistances $R_{1}$ and $R_{2}$ extends to infinity toward the right. Prove that the total resistance of the infinite network is equal to $R_{T} = R_{1} + \sqrt{R_{1}^{2} + 2 R_{1} R_{2}}$
(Hint: Since the network is infinite, the resistance of the network to the right of points c and d is also equal to $R_{T}$ .)

Short Answer

Expert verified

Proved that $R_{T} = R_{1} + \sqrt{R_{1}^{2} + 2 R_{1} R_{2}}$

Step by step solution

Equivalent/Total resistance of resistors in series and parallel

Equivalent/Total resistance of two resistors and in series is given as:

$R_{e q} = R_{1} + R_{2}$

Equivalent/Total resistance of two resistors and in parallel is given as:
$\frac{1}{R_{e q}} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$

Determine the total resistance of the resistors

Consider the following infinite resistors network of resistances $R_{1}$ and $R_{2}$ extends to infinity toward the right. Since the network is infinite, the resistance of the network to the right of points and is also equal to $R_{T}$ , as shown in the lower figure,

So, there are two parallel resistors $R_{2}$ and $R_{T}$ which are connected in series with two resistors of $R_{1}$

Thus,

$R_{T} = 2 R_{1} + {(\frac{1}{R_{2}} + \frac{1}{R_{T}})}^{- 1} R_{T} = 2 R_{1} + \frac{R_{2} R_{T}}{R_{2} + R_{T}} R_{T}^{2} - 2 R_{1} R_{T} - 2 R_{1} R_{2} = 0$