• hopkinmn
In summary, the equivalent resistance of an infinite resistive ladder made up of identical resistors R, measured between points A and B, is equal to (1+sqrt(3))*R, which can be found by adding one more ladder stage and solving for the equivalent resistance. Attempting to find the equivalent resistance by making all the parallel resistors into one equivalent resistor and dealing with infinities is not a valid approach.
hopkinmn

Homework Statement

Identical resistors, R, make up the legs and rungs of a resistive ladder of infinite height. Find Req (equivalent resistance) of ladder, measured between points A and B.

The Attempt at a Solution

Here's my thought process:
First make all the parallel resistors (the rungs) into one equivalent resistor. Since it's infinitely tall, this would be R^infinity/(infinity*R)
Second, find the total Req by adding the equivalent resistance for the rungs to the sum of the resistors on the legs. This would make Req=R^infinity/(infinity*R)+(infinity*R)

This makes Req equal to infinity.

Is my process for setting this up wrong?

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hopkinmn said:

Homework Statement

Identical resistors, R, make up the legs and rungs of a resistive ladder of infinite height. Find Req (equivalent resistance) of ladder, measured between points A and B.

The Attempt at a Solution

Here's my thought process:
First make all the parallel resistors (the rungs) into one equivalent resistor. Since it's infinitely tall, this would be R^infinity/(infinity*R)
Second, find the total Req by adding the equivalent resistance for the rungs to the sum of the resistors on the legs. This would make Req=R^infinity/(infinity*R)+(infinity*R)

This makes Req equal to infinity.

Is my process for setting this up wrong?

It's at best specious What justification do you have for "making all the parallel resistors (the rungs) into one equivalent resistor"? They are clearly not in parallel since their leads are not all directly connected to each other.

Also, doing math with infinities is not a straightforward process. Best avoided!

Instead, consider what would happen to Req if you were to add an additional ladder stage at AB (thus creating a new bottom end, say A' B'). How would the equivalent resistance of the infinite ladder be affected?

Also, I'm not sure I understand what you meant by :
gneill said:
They are clearly not in parallel since their leads are not all directly connected to each other.
So if the rungs were in parallel, there wouldn't be any resistors in the legs (above the first rung)?

hopkinmn said:
By how much would you expect the total to increase if you add 1 to infinity?

The ladder with equivalent resistance Req already has an infinite number of ladder segments. What do you suppose will happen if one more is added?
Also, I'm not sure I understand what you meant by :
They are clearly not in parallel since their leads are not all directly connected to each other.
So if the rungs were in parallel, there wouldn't be any resistors in the legs (above the first rung)?

Components are in parallel if they share exactly two nodes for their connections. When I look at that ladder I see a lot more than two nodes!

If you added one more, then the Req would just stay the same, since there's already an infinite amount.

hopkinmn said:
If you added one more, then the Req would just stay the same, since there's already an infinite amount.

Yup. So draw the circuit with the current ladder represented as Req, and add another ladder stage. Solve for the 'new' equivalent resistance...

So I'm still confused how to find the "original" Req.

To find the "new" Req, would you set it up so the old Req and the resistor in the rung are in parallel? And then add the two R.

hopkinmn said:
So I'm still confused how to find the "original" Req.
The original Req is "Req". It's a variable representing the currently unknown value.
To find the "new" Req, would you set it up so the old Req and the resistor in the rung are in parallel? And then add the two R.

Yup.

gneill said:
The original Req is "Req". It's a variable representing the currently unknown value.

But is there a way to find that variable in terms of just R?

hopkinmn said:
But is there a way to find that variable in terms of just R?

That's what you're doing by finding the equivalent resistance with one more stage tacked on.

Ok, I see now, thank you

1. What is a resistive ladder of infinite height?

A resistive ladder of infinite height is a theoretical circuit made up of an infinite number of resistors connected in series. This means that the resistors are connected one after the other, with no branching or parallel connections. The resulting circuit has a progressively higher resistance as you move along the ladder, hence the name "infinite height".

2. What is the purpose of a resistive ladder of infinite height?

The resistive ladder of infinite height is often used as a simplified model for analyzing more complex circuits. It allows scientists to make approximations and predictions about the behavior of circuits without having to deal with a large number of individual components.

3. How does the resistance change in a resistive ladder of infinite height?

In a resistive ladder of infinite height, the resistance increases in a linear fashion. This means that each resistor in the ladder has a value that is equal to the sum of all the resistors before it, resulting in a constantly increasing resistance as you move along the ladder.

4. What is the voltage distribution in a resistive ladder of infinite height?

The voltage in a resistive ladder of infinite height is also distributed in a linear fashion. This means that the voltage drops across each resistor are equal, resulting in a constant decrease in voltage as you move along the ladder. This voltage distribution is important in understanding how the current flows through the circuit.

5. Can a resistive ladder of infinite height be built in real life?

No, a resistive ladder of infinite height cannot be physically built due to the infinite number of resistors required. However, it can be simulated using computer software or approximated using a large number of resistors in a circuit. The concept of an infinite ladder is used mainly for theoretical purposes in the field of circuit analysis.

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