An infinite ladder of resistors

Problem 3.152 of Irodov

The figure below shows an inifite circuit formed by the repetition of the same link, consisting of resistance R_1 = 4.0 \ \Omega and R_2 = 3.0 \ \Omega. Find the resistance of this circuit between points A and B.

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Let’s denote the resistance between the points A and B by R_{AB}. Since the circuit is infinite, removing the first R_1 and R_2 resistors gives the same arrangement back again — the arrangement is self-similar. That means, the resistance between the points C and D is just R_{AB} without the left-most R_1 and R_2 resistors, and we may redraw the circuit as shown below.

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It is now straightforward to calculate the resistance,

(1)   \begin{equation*}   R_{AB} = R_1 + \frac{R_2 R_{AB}}{R_2 + R_{AB}} \end{equation*}

which gives a quadratic equation for R_{AB},

(2)   \begin{equation*}   R_{AB}^2 - R_1 R_{AB} - R_1 R_2 = 0 . \end{equation*}

This equation has two roots. We keep only the positive root, because resistance cannot be a negative number.

(3)   \begin{equation*}   \boxed{     R_{AB}       % = \frac{R_1 \pm \sqrt{R_1^2 + 4 R_1 R_2}}{2}       = \frac{R_1}{2} \left( 1 + \sqrt{ 1 + \frac{4 R_2}{R_1} } \right)     } . \end{equation*}

Plugging in R_1 = 4.0 \ \Omega and R_2 = 3.0 \ \Omega, we find R_{AB} = 6.0 \ \Omega.

Bonus Problem: At what value of the resistance R_x in the circuit shown below will the total resistance between points A and B be independent of the number of cells. (Hint: What value of R_x will make the circuit appear self-similar at the lower rungs of the ladder?)

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Answer: (\sqrt{3}-1)R.

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