Question

26.83 An Infinite Network. As shown in Fig. P26.83, a network of resistors of resistances ${\mathit{R}}_{\mathbf{1}}$and${\mathit{R}}_{\mathbf{2}}$extends to infinity toward the right. Prove that the total resistance of the infinite network is equal to${\mathit{R}}_{\mathbf{T}}\mathbf{=}{\mathit{R}}_{\mathbf{1}}\mathbf{+}\sqrt{{\mathbf{R}}_{\mathbf{1}}^{\mathbf{2}}\mathbf{+}\mathbf{2}{\mathbf{R}}_{\mathbf{1}}{\mathbf{R}}_{\mathbf{2}}}$

(Hint: Since the network is infinite, the resistance of the network to the right of points c and d is also equal to ${\mathit{R}}_{\mathbf{T}}$.)

Short answer

Proved that$R_T=R_1+\sqrt{R_1^2+2R_1{R}_2}$

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:

${\mathit{R}}_{\mathbf{e}\mathbf{q}}\mathbf{=}{\mathit{R}}_{\mathbf{1}}\mathbf{+}{\mathit{R}}_{\mathbf{2}}$

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

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,

$$\begin{gathered} R_T=2R_1+{\left(\frac{1}{R_2}+\frac{1}{R_T}\right)}^{-1} \\ {R}_T=2R_1+\frac{R_2{R}_T}{R_2+R_T} \\ {R}_T^2-2R_1{R}_T-2R_1{R}_2=0 \end{gathered}$$

This is a quadratic equation of$R_T$which has a solution of

$R_T=R_1\pm \sqrt{R_1^2+2R_1{R}_2}$

But since , then

$R_T=R_1+\sqrt{R_1^2+2R_1{R}_2}$

Hence, proved.