# Effective Resistance of an Infinitely Long Ladder of Resistors

Find the effective resistance (resistance between a and b) of an infinitely long ladder of resistors, as shown in the figure, each having resistance R.

http://people.tamu.edu/~malindenmoyer/fig.jpg [Broken]

The point is that the input resistance which we do not yet know--call it R--will not be changed by adding a new set of resistors to the front end of the chain to make it one unit longer. But now, adding this section, we see that this new input resistance is just R1 in series with the parallel combination of R2 and R. We get immediately an equation that can be solved for R.

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rl.bhat
Homework Helper
If you consider Req. at a'b' line, then R and Req are in parallel and other 2 R are in series with them. Simplify the circuit and solve the quadratic to find Req..

still confused, perhaps a more detailed hint?

collinsmark
Homework Helper
Gold Member
Hello malindenmoyer,

A very important part of this is your quote,

"[Req] will not be changed by adding a new set of resistors to the front end of the chain to make it one unit longer."

So take Req as the equivalent resistance looking to the right of line a'b'.

Model Req as a single resistor if you want, at this point. Now find the equivalent resistance once you add in the extra three R resistors, between points ab and a'b'. Find the equivalent resistance of that (you'll have 4 resistors total. 2 series R, in series with an R and an Req in parallel).

But you know from your above quote, that the equivalent resistance of that new circuit (above paragraph) won't change the equivalent resistance before the 3 new resistors were added. So you can set the whole thing equal to Req. Then solve for Req.

Okay, from both of the responses, I was able to come up with this expression, am I on the right track?

$$R_{eq}^2-\frac{2R^2-1}{R}R_{eq}-1=0$$

If correct, do I use the quadratic formula to solve?

collinsmark
Homework Helper
Gold Member
Okay, from both of the responses, I was able to come up with this expression, am I on the right track?

$$R_{eq}^2-\frac{2R^2-1}{R}R_{eq}-1=0$$

If correct, do I use the quadratic formula to solve?
Yes, you will be using the quadratic equation/formula for this one. But I don't think your equation is correct.

Hint: Each term in the equation (in the form aReq2 + bReq +c = 0) needs to have units of $$\Omega ^2$$ (meaning 'a' is unitless, 'b' has units of $$\Omega$$, and 'c' has units of $$\Omega^2$$).

rl.bhat
Homework Helper
Req = 2R + [ R*Req/(R + Req).]
Now simplify and solve for Req.