(a) Find an expression for the magnetic energy stored per unit length in a coaxial transmission line consisting of conducting sleeves of negligible thickness, having radii \(a\) and \(b\). A medium of relative permeability \(\mu_{r}\) fills the region between conductors. Assume current \(I\) flows in both conductors in opposite directions. (b) Obtain the inductance, \(L\), per unit length of line by equating the energy to \((1 / 2) L I^{2}\).

Ampere's Law plays a critical role in magnetic field calculations. It states that the integrated magnetic field around a closed path is proportional to the electric current passing through any surface bounded by the path. In the context of a coaxial cable, this law can be utilized to deduce the magnetic field present due to a steady current flowing along the conductors.

The mathematical expression of Ampere's Law for our coaxial line system is given by:
\[\begin{equation} B = \frac{\mu_r \mu_{0} I}{2\pi r}\end{equation}\]where B represents the magnetic field, \(I\) is the current, \(\mu_0\) is the permeability of free space, \(\mu_r\) is the relative permeability of the medium, and \(r\) is the radial distance from the wire. This equation is pivotal for understanding the magnetic field distribution in the region between the two coaxial conductors which ultimately influences the transmission line's energy and inductive characteristics.

The calculation of the magnetic field within a coaxial transmission line involves understanding the spatial distribution of magnetic flux density. From Ampere's Law, we identify that the magnetic field is inversely proportional to the radial distance, meaning it decreases as one moves away from the central conductor.

The coaxial line scenario presents a straightforward implementation of Ampere's Law yielding the formula:
\[\begin{equation} B = \frac{\mu_r \mu_{0} I}{2\pi r}.\end{equation}\]This formula can be comprehensively applied between the conductors' radii a and b to determine the magnetic field at any point. A key revelation from this calculation is that the magnetic field is singularly dependent on the radial position, simplifying the approach to determining the magnetic field's influence on the energy stored in the coaxial cable.

Energy density is a measure of the energy stored in a particular field per unit volume. For magnetic fields, the formula to calculate the energy density, denoted as \(u_m\), is:
\[\begin{equation} u_m = \frac{1}{2\mu_r\mu_0}B^2.\end{equation}\]This indicates how much energy is stored in the magnetic field per unit volume. The calculation of the total energy stored in the coaxial transmission line involves integrating this energy density over the space between the conductors. Given that the conductors are cylindrical and the field's radial dependence, the integration is carried out across the radius from a to b. This integrated energy density gives us a complete picture of the total magnetic energy per unit length in a coaxial transmission line.