# Capacitance of Coaxial Cable Configuration with Multiple Conductors

• DRussian
In summary, the problem involves a cable with three coaxial conductors - a solid inner cylinder and two thin cylindrical shields. The space between the conductors is filled with air and the inner cylinder and outer shield are connected at one end. The question asks for the capacitance per unit length of this configuration. The attempt at a solution involves using Gauss's Law to find the change in voltage from the inner cylinder to the first inner shell. However, the questioner is confused about how to account for the second, outer shell and the direction of the electric field in the two different spaces.
DRussian

## Homework Statement

I can't seem to understand this problem (I do see that the inner cylinder and outer shell will have the same charge, but I can't see what I'm supposed to assume about the middle shell):

A cable consists of three conductors, a solid inner cylinder and two thin cylindrical shields. All three conductors are coaxial, with the solid inner conductor, central shield, and outer shield having diameters 0.55 mm, 3.43 mm, and 5.88 mm, respectively. Assume the space between the conductors is filled with air. The inner cylinder and outer shield are connected at one end of the cable.

What is the capacitance per unit length of this configuration?

ΔV = ∫E ds
C = Q/(ΔV)

## The Attempt at a Solution

I tried finding the resulting change in voltage from the inner cylinder to the first inner shell.

Solving from Gauss's Law, I first got that the magnitude of the electric field of a cylindrical charge distribution having linear charge density λ is (k is Boltzman's constant) E = 2kλ/r.
Integrating from the radius of the center to the first inner shell, I get:

ΔV = ∫E ds = -2kλ*ln(b/a) (where b = radius from center of capacitor to first inner shell and a = radius of inner solid conductor).

This is where I got confused; how would I deal with the second, outer shell into here? How would I account that the E-field is in the opposite direction in the second space (between the two hollow shells) as in the first space (between the solid conductor and first shell)?

DRussian said:
I do see that the inner cylinder and outer shell will have the same charge,
No, the cylindrical core and the outermost shield will not have the same charge. What is it that will be the same for them?

## 1. What is a coaxial capacitor?

A coaxial capacitor is a type of capacitor that consists of two conductive plates separated by a dielectric material, with one plate nestled inside the other like a coaxial cable. It is used to store and release electrical energy in electronic circuits.

## 2. What is the problem with coaxial capacitors?

The most common problem with coaxial capacitors is the formation of a dielectric breakdown, which can occur when the voltage across the capacitor exceeds its breakdown voltage. This can lead to a short circuit and potential damage to the circuit or equipment.

## 3. How can I prevent dielectric breakdown in a coaxial capacitor?

To prevent dielectric breakdown, it is important to choose a capacitor with a breakdown voltage that is higher than the expected operating voltage of the circuit. In addition, proper insulation and spacing between the conductive plates can help prevent breakdown.

## 4. How do I test for a faulty coaxial capacitor?

To test for a faulty coaxial capacitor, you can use a multimeter to measure the capacitance and resistance of the capacitor. If the measured values deviate significantly from the expected values, there may be a problem with the capacitor. You can also visually inspect the capacitor for any signs of damage or leakage.

## 5. Can a faulty coaxial capacitor be repaired?

In most cases, a faulty coaxial capacitor cannot be repaired and will need to be replaced. Attempting to repair a faulty capacitor can be dangerous and may further damage the circuit or equipment. It is best to consult a professional or replace the capacitor with a new one.

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