Section 3.2 · Worked examples · 1 of 2
Worked example 3.2.1: Air-spaced coaxial line
Two short calculations tied to the applications. Each one is Gauss's law plus one symmetry argument, and nothing else is needed. The numerical constants used are , Eq. (1.4.2), , and . A good way to use them: read the given quantities, choose the Gaussian surface and write yourself, and only then put the numbers in.
Air-spaced coaxial line
#cf. Problems 3.2.1–3.2.2
Given. An air-insulated coaxial line has an inner conductor of radius and a screen of inner radius . The inner conductor carries , and the screen is neutral.
Find. (a) at and , (b) on the inner conductor and on the inner face of the screen, and (c) the largest if air breaks down at .
A coaxial Gaussian cylinder of radius between the conductors encloses only the charge of the inner conductor, so the field there is that of Problem 3.2.1, for (Figure 3.2.15). Inside the metal of the screen , so a Gaussian cylinder drawn in the metal encloses zero charge, which fixes the charge on the inner face of the screen.
At the surface of the inner conductor and at the screen,
since .
The charge per unit length is spread over the circumference of each surface. A Gaussian cylinder inside the screen metal encloses , so the inner face of the screen carries per metre:
Check: at a conductor surface, and .
The field is largest at , so breakdown begins there. Setting ,
The inner conductor sees times the field at the screen, so the insulation fails there first. The given line could carry about eight times its present charge. Because the screen is neutral, it carries on its inner face and on its outer face, and outside the cable there is the field of a line charge . If the screen is earthed, the outer charge flows away and no field at all leaves the cable (Application 3.2.2).
Set the radii and the line charge below, and switch the earthing of the screen on and off: the field peaks at the inner conductor and, with the screen earthed, vanishes outside.