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Attenuator Chains and Axons. The infinite network of resistors shown in Fig. 26.83 is known as an attenuator chain, since this chain of resistors causes the potential difference between the upper and lower wires to decrease, or attenuate, along the length of the chain. a) Show that if the potential difference between the points \(a\) and \(b\) in Fig. 26.83 is \(V_{a b}\) , then the potential difference between points \(c\) and \(d\) is \(V_{e d}=V_{a b} /(1+\beta),\) where \(\beta=2 R_{1}\left(R_{T}+R_{2}\right) / R_{T} R_{2}\) and \(R_{T},\) the total resistance of the network, is given in Challenge Problem \(26.91 .\) (See the hint given in that problem.) (b) If the potential difference between terminals \(a\) and \(b\) at the left end of the infinite net- work is \(V_{0}\) , show that the potential difference between the upper and lower wires \(n\) segments from the left end i \(V_{n}=V_{0} /(1+\beta)^{n}\) . If \(R_{1}=R_{2},\) how many segments are needed to decrease the potential difference \(V_{n}\) to less than 1.0\(\%\) of \(V_{0} ?(\mathrm{c})\) An infinite attenuator chain provides a model of the propagation of a nvoltage pulse along a nerve floer, or axon. Each segment of the network in Fig. 26.83 represents a short scgment of the axon of length \(\Delta x\) . The resistors \(R_{1}\) represent the resistance of the fluid inside and \(R_{2}=8.0 \times 10^{8} \Omega\) (the membrane wall is a good insulator).

Step by step solution

01

Understand the Problem

The problem involves an infinite network of resistors that attenuate the potential difference between two points. We need to compute the potential drop along the chain using given parameters.

02

Derive the Expression for Potential Difference Vcd

Consider one section of the network, which is a T-section with resistors \( R_1 \), \( R_2 \), and connected to another section. Let the total equivalent resistance of the chain be \( R_T \). Using basic circuit analysis principles, simplify the circuit to derive the relation \( V_{cd} = \frac{V_{ab}}{1 + \beta} \), where \( \beta = \frac{2R_1 (R_T + R_2)}{R_TR_2} \).

03

Verify Expression for Infinite Chain

Insert the expression of \( R_T \) from the problem into the equation to confirm that the attenuation equation is consistent with the infinite extension of the chain.

04

Calculate Potential Difference after n Segments

Using the derived expression for one segment, apply it recursively for \( n \) segments. The potential difference \( V_n = \frac{V_0}{(1 + \beta)^n} \) involves exponentiation due to repeated attenuation.

05

Determine Number of Segments for 1% Potential

Given \( R_1 = R_2 \) and \( R_T \), calculate \( \beta \). Set \( V_n < 0.01V_0 \). Solve \( \frac{1}{(1+\beta)^n} < 0.01 \) to find \( n \). Utilize logarithms to solve for \( n \).

06

Analyze Axon Model

Interpret the attenuator chain as a model for a nerve cell's axon, where each segment represents a length \( \Delta x \). Resistors \( R_1 \) model the intracellular fluid resistance, and \( R_2 = 8.0 \times 10^8 \Omega \) represents the insulation provided by the axon membrane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resistor Networks

Resistor networks play a crucial role in understanding electric circuits. These networks consist of multiple resistors connected in various configurations, such as series, parallel, or a combination of both. The primary purpose of resistor networks is to control the flow of electric current and adjust the potential differences across various parts of a circuit. In the context of the attenuator chain, each network segment consists of resistors that influence the voltage drop along the chain.
In such networks, calculations often focus on determining the equivalent resistance, which simplifies analysis by reducing complex networks to a single resistor. This is achieved using formulas specific to series and parallel configurations:

• Series connection: The total resistance is the sum of individual resistances, i.e., \( R_{total} = R_1 + R_2 + \ldots + R_n \).
• Parallel connection: The reciprocal of the total resistance is the sum of the reciprocals of individual resistances, i.e., \( \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} \).

This foundation allows for complex networks like the infinite attenuator chain to be analyzed and simplified, providing insight into the potential differences across the network.

Potential Difference

The concept of potential difference, often referred to as voltage, is fundamental to the study of electric circuits. It is the driving force that moves electric charges through a circuit, measured in volts. In simple terms, the potential difference between two points is the work needed per unit charge to move a charge between these points.
A higher potential difference between two points indicates a stronger force to drive electric current from one point to another. In the attenuator chain, the purpose is to reduce or "attenuate" this potential difference progressively along the chain. The formula \( V_{cd} = \frac{V_{ab}}{1 + \beta} \) shows how this reduction happens across each segment of the resistor chain. This iterative process effectively reduces the voltage across sections of the network, which is critical for understanding real-world applications like signal weakening over long distances in a circuit.

Nerve Axons

Nerve axons are an integral component of the human nervous system. These long, slender projections of nerve cells transmit electrical impulses away from the nerve cell body. The propagation of electrical signals along axons is remarkably similar to the process in resistor networks.
The infinite attenuator chain provides a model for signal propagation along nerve axons. Here, electrical resistance inside the axon and insulation by the axon membrane can be represented using resistors, like \( R_1 \) for intracellular fluid resistance and \( R_2 = 8.0 \times 10^8 \Omega \) for axon membrane insulation.
This analogy helps to understand how nerve signals can diminish over distance, unless regenerated by biological processes. It highlights the importance of resistive elements in electrical and biological systems, ensuring efficient transmission and processing of signals across long distances.

Circuit Analysis

Circuit analysis is a vital skill for anyone looking to understand or design electronic circuits. It involves deducing unknown quantities like current, voltage, and resistance in any part of a circuit. For complex configurations, systematic methods such as the node-voltage method or mesh current analysis are employed.
In the context of the attenuator chain, circuit analysis involves breaking down the network into manageable sections and using principles like Kirchhoff's laws to determine potential differences across terminals. For example, determining \( V_{cd} = \frac{V_{ab}}{1 + \beta} \) is part of this analysis which relies on correctly calculating the total resistance, \( R_T \), and other vital parameters.
Repeated attenuation, explored using recursive formulas, helps predict behavior over an extended number of segments. These processes, coupled with tools like LaTeX for mathematical precision, make circuit analysis a powerful approach for engineering and scientific inquiries into electronic behavior.