Find the equivalent model of an infinite ladder circuit

[The Blind Watchmaker]

Homework Statement

2. Homework Equations
Ohm's law and Kirchoff's voltage law

The Attempt at a Solution

My solution is a bit long so I will just briefly explain it. First, we find the total equivalent resistance. Since the circuit extends to infinity, it is equal to replacing the second branch onward by a single resistor. After some calculus, Req = 2r. Thus, internal resistance is 2r. Next, apply the loop rule to find the current before it terminates as it is physically impossible for the current to blow up to infinity. Effective emf is found to be 2Ir = 2ε.

My question is, is there perhaps another (shorter) way to solve this problem? If so, please explain your solution. Thanks!

 

[ehild]

Homework Statement

View attachment 221489

2. Homework Equations
Ohm's law and Kirchoff's voltage law

The Attempt at a Solution

My solution is a bit long so I will just briefly explain it. First, we find the total equivalent resistance. Since the circuit extends to infinity, it is equal to replacing the second branch onward by a single resistor. After some calculus, Req = 2r. Thus, internal resistance is 2r. Next, apply the loop rule to find the current before it terminates as it is physically impossible for the current to blow up to infinity. Effective emf is found to be 2Ir = 2ε.

My question is, is there perhaps another (shorter) way to solve this problem? If so, please explain your solution. Thanks!

Your method and your results are correct. You can use Thevenin equivalent. Adding the unit in the red frame to the equivalent source of emf E and internal resistance Ri, the emf and internal resistance of the Thevenin equivalent between A and B are E and Ri, respectively.

 

[scottdave]

Can you show what you mean by replacing the "2nd branch" with a resistor? Perhaps post a picture.

 

[scottdave]

You can use Thevenin equivalent. Adding the unit in the red frame to the equivalent source of emf E and internal resistance Ri,...

View attachment 221497

Now I'll buy that. I appeared as if the OP was going to replace everything beyond the first source with just a resistor.