An infinite ladder of resistors

In summary, the network of resistors shown extends off to infinity, with equal resistors RS on the sides and different resistors Rc in the center. The general expressions for the resistance RT between points a and b and the potential difference between points c and d in terms of Rs, Rc, RT, and the potential difference Vab between points a and b can be found using a recursion relationship. This is due to the infinite arrangement of the ladder, where the remaining ladder after the c-d branch can be replaced with an equivalent resistance R. It is suggested to work out the resistance for finite ladders to determine a general formula and apply a limit.
  • #1
phys121
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The network of resistors shown extends off to infinity. The resistors on the
sides of the “ladder”, RS, are equal to each other,but different from the resistors in the center of the ladder,Rc

.....Rs........Rs.....Rs
a------^^^^^------c---------^^^^^------------^^^^^------------etc.
......>......>......>
......<......<......<
......>Rc......>Rc.....>Rc
......<......<......<
......>......>......>
b------^^^^^------d---------^^^^^------------^^^^^------------etc.
.....Rs......Rs......Rs

1) find a general expression for the resistance RT between points a and b.

2) find a general expression for the potential difference between points c and d in
terms of the Rs , Rc , Rt , and the potential difference between points a and b, Vab . Any Help would be very much appreciated. have been sitting her for many hours now just stuck!

Ps. Please ignore all the full stops, it was the only way to make the circuit look like it should. So they actually mean nothing.
Between the two 'ladder' sides there is a resistor Rc just in case its not 100% clear. thanks in advance
 
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  • #2
Well this could have been a classic sum had been an infinite maze. But its quite easy as it s a ladder. The thing you have to note is that this is an infinite arangement.Lets say that the resisatnce betwen a and b is R. Then you can assume that after the c d branch you can replace the remaining ladder with R.
 
  • #3
I suggest you work out the resistance for a couple of finite ladders to see if there is a general formula to which you can apply a limit.
 
  • #4
On second thought you can use a recursion relationship. Remember that the whole ladder is one rung plus another identical infinite ladder.
 

1. What is an infinite ladder of resistors?

An infinite ladder of resistors is a theoretical circuit made up of an infinite number of identical resistors connected in a specific pattern. This circuit is used to model and understand the behavior of more complex circuits.

2. How does an infinite ladder of resistors work?

The infinite ladder of resistors works by dividing the voltage across it evenly among each resistor in the circuit. This allows for an even distribution of the current and helps to regulate the flow of electricity.

3. What are the advantages of using an infinite ladder of resistors?

One advantage of using an infinite ladder of resistors is that it can simplify the analysis of more complex circuits. It can also help in understanding the behavior of circuits with a large number of resistors.

4. What are the limitations of an infinite ladder of resistors?

One limitation of an infinite ladder of resistors is that it is a theoretical concept and cannot be physically constructed. It also assumes ideal conditions, which may not always be present in real-life circuits.

5. How is an infinite ladder of resistors used in practical applications?

An infinite ladder of resistors is not used in practical applications, but the concept is applied in the design and analysis of more complex circuits. It can also be used as a teaching tool to understand the behavior of circuits with a large number of resistors.

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