Pairs of parallel wires or coaxial cables are two conductors separated by an insulator, so they have a capacitance. For a given cable, the capacitance is independent of the length if the cable is very long. A typical circuit model of a cable is shown in the figure below. It is called a lumped-parameter model and represents how a unit length of the cable behaves. Find the equivalent capacitance of the following lengths. (Use the following as necessary: C.) (a) one unit length (Figure A)C1 = (b) two unit lengths (Figure B)C2 = (c) an infinite number of unit lengths (Figure C). Hint: For the infinite number of units, adding one more unit at the beginning does not change the equivalent capacitance.C∞ = ### Understanding Capacitance in Parallel Wires and Coaxial CablesPairs of parallel wires or coaxial cables act as two conductors separated by an insulator, thereby exhibiting capacitance. For a given cable, the capacitance remains independent of the length if the cable length approaches infinity. A typical circuit model representing how a unit length of the cable behaves is illustrated below. This model is known as a lumped-parameter model.#### Capacitance Modeling **A. One unit length** ![One Unit Length](#) \[ C_1 = \frac{CC}{C + C} \]- In Figure A, the single unit length of the cable is modeled with two capacitors \( C \) in series, resulting in an equivalent capacitance \( C_1 \). **B. Two unit lengths** ![Two Unit Lengths](#) \[ C_2 = \frac{CC}{C + C} \]- In Figure B, two unit lengths of the cable are considered. Although two unit lengths are modeled, the equation given under (b) is incorrect. A proper series and parallel combination analysis should be conducted for an accurate equivalent capacitance. **C. Infinite number of unit lengths** ![Infinite Unit Lengths](#) \[ C_{\infty} = 0.618C \]- In Figure C, an infinite number of unit lengths are represented by a repeating pattern of capacitors \( C \). For an infinite series of unit lengths, adding one more unit at the beginning does not alter the equivalent capacitance. The equivalent capacitance for an infinite number of units, \( C_{\infty} \), is approximately \( 0.618C \), though further verification is required since the equation under (c) is marked as incorrect.### SummaryUnderstanding the equivalent capacitance in different cable lengths aids in the design and analysis of electrical systems where such configurations are in use. Here's how you can compute the capacitance for one unit length, two unit lengths (with correct calculations), and an indefinite length systematically.--- For further reading, ensure you reference your circuit theory textbooks or reliable resources to verify complex calculations.

The capacitance is independent of the length if the cable is very long. The lumped parameter model…

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College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Pairs of parallel wires or coaxial cables are two conductors separated by an insulator, so they have a capacitance. For a given cable, the capacitance is independent of the length if the cable is very long. A typical circuit model of a cable is shown in the figure below. It is called a lumped-parameter model and represents how a unit length of the cable behaves. Find the equivalent capacitance of the following lengths. (Use the following as necessary: C.)

(a) one unit length (Figure A)

C₁ =

(b) two unit lengths (Figure B)

C₂ =

(c) an infinite number of unit lengths (Figure C). Hint: For the infinite number of units, adding one more unit at the beginning does not change the equivalent capacitance.

C_∞ =

### Understanding Capacitance in Parallel Wires and Coaxial Cables

Pairs of parallel wires or coaxial cables act as two conductors separated by an insulator, thereby exhibiting capacitance. For a given cable, the capacitance remains independent of the length if the cable length approaches infinity. A typical circuit model representing how a unit length of the cable behaves is illustrated below. This model is known as a lumped-parameter model.

#### Capacitance Modeling

<div style="text-align: center;">
**A. One unit length**
![One Unit Length](#)
</div>

\[ C_1 = \frac{CC}{C + C} \]

- In Figure A, the single unit length of the cable is modeled with two capacitors \( C \) in series, resulting in an equivalent capacitance \( C_1 \).

<div style="text-align: center;">
**B. Two unit lengths**
![Two Unit Lengths](#)
</div>

\[ C_2 = \frac{CC}{C + C} \]

- In Figure B, two unit lengths of the cable are considered. Although two unit lengths are modeled, the equation given under (b) is incorrect. A proper series and parallel combination analysis should be conducted for an accurate equivalent capacitance.

<div style="text-align: center;">
**C. Infinite number of unit lengths**
![Infinite Unit Lengths](#)
</div>

\[ C_{\infty} = 0.618C \]

- In Figure C, an infinite number of unit lengths are represented by a repeating pattern of capacitors \( C \). For an infinite series of unit lengths, adding one more unit at the beginning does not alter the equivalent capacitance. The equivalent capacitance for an infinite number of units, \( C_{\infty} \), is approximately \( 0.618C \), though further verification is required since the equation under (c) is marked as incorrect.

### Summary

Understanding the equivalent capacitance in different cable lengths aids in the design and analysis of electrical systems where such configurations are in use. Here's how you can compute the capacitance for one unit length, two unit lengths (with correct calculations), and an indefinite length systematically.

---
For further reading, ensure you reference your circuit theory textbooks or reliable resources to verify complex calculations.

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