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  4. An infinite ladder network is arranged with resistances R and 2R as shown. The effective resistance between terminals A and B is

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Q. An infinite ladder network is arranged with resistances $ R $ and $ 2R $ as shown. The effective resistance between terminals A and B isPhysics Question Image

AMUAMU 1999

Solution:

: Let the total resistance of the infinite ladder network be Z. Subtract first step from the infinite network. The total resistance of the remaining ladder still remains to be Z. $ \therefore $ Resistance between C and $ D=\frac{2R\times Z}{2R+Z} $ Total resistance between A and $ B=R+\frac{2RZ}{2R+Z} $
$ \therefore $ $ Z=R+\frac{2RZ}{2R+Z} $ Or $ (2RZ+{{Z}^{2}}) $ $ =(2{{R}^{2}}+2Z)+2RZ $ or $ {{Z}^{2}}-RZ-2{{R}^{2}}=0 $ or $ (Z-2R)(Z+R)=0 $ or $ Z-2R=0; $ or $ Z=2R $

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