Imagine a circuit like the one above, all with identical resistors with individu

Question

Imagine a circuit like the one above, all with identical resistors with individual resistance R The equivalent resistance across points a and b is defined as where delta V_ab is a hypothetical potential difference between the points and / is the resulting current. Since this circuit can be reduced to "in series" and "in parallel" parts, the equivalent resistance of this circuit is Now imagine that we extend the circuit in the following manner Now the equivalent resistance of this circuit is Show that if we extend the circuit in the same manner infinitely. the equivalent resistance R_ab will coverge to Note: YOU Jon 7 need to do show a rigorous epsilon-delta proo//or convergence You may use the informal (hut approximately valid) argument that i/R^ converges to some value, then extending the circuit one more time should result in the same value for

Explanation / Answer

in First case given

Rab = R + 1/[1/R + 1/2R]

now in second case

Rab = R + 1/[1/R + [1/[R + 1/[1/R + 1/2R]]]

now in case same sequence infinite time the bold part will repeat always

So use eq 1

[R + 1/[1/R + 1/2R] = Rab

And put this into eq 2

Rab = R + 1/[1/R + 1/Rab]

Rab = R + [R*Rab/[Rab + R]]

Rab = (R*Rab + R^2 + R*Rab)/[Rab + R]

Rab^2 + R*Rab = (R*Rab + R^2 + R*Rab)

Rab^2 - R*Rab - R^2 = 0

Rab = x

x^2 - Rx + R^2 = 0

Solve this equation

x = [R + sqrt(R^2 - 4*(-R^2)*1)]/2*1

x = Rab = R*[1 + sqrt 5]/2

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