Question

A coaxial transmission line has a solid inner conductor of radius a and a thin-walled outer conductor of radius b. The medium between the conductors has permeability μ_0. Find the inductance per meter of length of the line:

(a) if all current of the inner conductor is on its surface (at radius a) (actually a thin layer of negligible thickness)

(b) if the current of the inner conductor is uniformly distributed throughout its cross-sectional area; that is, if the internal inductance of the inner conductor is taken into account.

Answer 1

The self-inductance per meter for a coaxial transmission line depends on the radius of the inner and outer conductors. For surface current distribution, the inductance is μ0/(2π) × In(b/a), while for uniform current distribution, the inductance per meter also includes an additional term from internal inductance.

Step-by-step explanation:

To find the inductance per meter for a coaxial transmission line with a solid inner conductor of radius a and a thin-walled outer conductor of radius b, and with the medium between the conductors having permeability μ0, we consider two cases based on the distribution of the current in the inner conductor.

Case (a) - Surface Current Distribution

If all current I of the inner conductor is on its surface, we use Ampère's law to find the magnetic field and then the magnetic energy per unit length. The self-inductance per unit length (L) is given by:

L = μ0/(2π) × In(b/a)

Case (b) - Uniform Current Distribution

If the current is uniformly distributed throughout the cross-sectional area of the inner conductor, we must include the internal inductance. In this case, the self-inductance per unit length is higher due to the additional internal inductance and is given by:

L = μ0/(2π) × In(b/a) + μ0/(8π)

Note that the additional term (μ0/(8π)) in case (b) represents the internal inductance contribution, which was zero in case (a) since we assumed the current was only on the surface.