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Question

Find the effective resistance between A & B.

A
7R12
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B
R(1+3)2
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C
R(1+5)2
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D
R12
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Solution

The correct option is C R(1+√5)2Let the effective resistance between A & B be RE. Since the network is infinite long, so removal of one group of resistance from the chain will not change the network. Therefore, effective resistance between points C & D would also be RE. The equivalent network will be as shown below. The original infinite chain is equivalent to R in series with R & RE in parallel. So the effective resistance of infinite network will be RE=R+RRER+RE ⇒RER+R2E=R2+2RRE ⇒R2E−RRE−R2=0 ∴RE=R±√R2−4⋅1⋅(−R2)2⋅1 ⇒RE=R(1±√5)2 Taking positive value, because resistance can't be negative. ∴RE=R(1+√5)2 Hence, option (a) is correct. Key Concept: In an infinitely long network, addition or removal of one section from the chain still leaves the chain infinite (i.e. just about the same as before).

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