Resistor Network: Find the Equivalent Resistance Across AB

[lingling]

A resistor network is built up indefinitely as shown in the figure. The equivalent resistance across AB is
A. 1 ohm
B. 1.24 ohm
C. 2 ohm
D. unable to be determined.

-> I don't know where to start with. Can anyone give me some hints?

 

[HallsofIvy]

Am I missing something? It looks to me like the picture says that the resistance across AB is 2 ohms. The rest of the circuit is irrelevant.

 

[Hayden]

Am I missing something? It looks to me like the picture says that the resistance across AB is 2 ohms. The rest of the circuit is irrelevant.

If you put one end of a battery at A and the other end at B, current will flow not just across the 2ohm resistor you speak of but all the other resistors aswell.

A quick calculation indicates the following.

The resistance for a network of length 1 is 4/3 ohms since 1/(1/2+1/4)=1/(3/4)=4/3

The next value obtained by extending the network by an extra 3 resistors is can be found by again just using the rules for parallel and series resistors. I found this to be equal to 5/4 ohms.

Hence the trend appears to be

4/3
5/4

I would be extremely surprised if this trend did not continue and the next value would be 6/5. In that case if the network is infinitely long the resistance would tend to 1ohm

 

[lingling]

But the answer is 1.24 ohm.
I can't understand. Why it is not 2 ohm...?

 

[doodle]

Interesting question...

Assume that the total resistance is R. The resistor network right-of (and including) the second 2ohm resistor from the left is exactly the same as the entire resistor network. Thus, we have
R = 2 || (2+R)
Solving for R gives you the desired answer.

 

[lingling]

I still cannot understand.
Is there any simpler approach?

 

[OlderDan]

I still cannot understand.
Is there any simpler approach?

doodle has the answer. There is no easier way. There is no end to the chain of resistors, and if you think about how the current will flow you will realize that each 2-ohm resistor will have less current than the previous 2-ohm resistor. If you removed all the resistors touching points A and B, you would have exactly the same net resistance you have with them there.

This problem is a precursor to the important concept of characteristic impedence of transmission lines.

http://www.allaboutcircuits.com/vol_2/chpt_13/3.html