The inner and outer dimensions of a coaxial copper transmission line are 2 and \(7 \mathrm{~mm}\), respectively. Both conductors have thicknesses much greater than \(\delta\). The dielectric is lossless and the operating frequency is \(400 \mathrm{MHz}\). Calculate the resistance per meter length of the \((a)\) inner conductor; (b) outer conductor; \((c)\) transmission line.

Electrical resistivity is a fundamental property that quantifies how strongly a material opposes the flow of electric current. The lower the resistivity, the better the material is at conducting electricity. For example, copper is widely used in electrical applications due to its low resistivity, typically around \(1.72 \times 10^{-8} \Omega \cdot m\).
Resistivity is affected by various factors, including temperature, material purity, and physical structure. In the context of coaxial transmission lines, the resistivity of the conductors dictates the resistance observed as current travels along the line. To find the resistance per meter length of a conductor, one must divide the resistivity by the conductor's cross-sectional area. This calculation provides an invaluable understanding of how the material's properties influence the overall performance of the transmission line.

The cross-sectional area is crucial for calculating the resistance of conductors in a transmission line. It plays an inverse role in determining resistance: a larger cross-sectional area leads to lower resistance, while a smaller area increases resistance.To calculate the cross-sectional area of cylindrical conductors, such as those in coaxial cables, the formula \( A = \pi r^2\) is employed, where \(r\) is the radius of the conductor. This calculation considers only the circular part of the conductor through which current flows. In the case of a coaxial cable's outer conductor, the area is the difference between the areas of two circles: the larger representing the outer dimension and the smaller representing the inner dimension immediately surrounding the dielectric.
Understanding these dimensions is essential when designing and analyzing circuits, as properly sized conductors are required to handle the expected current without excessive heat or power loss.

Transmission lines, such as coaxial cables, exhibit frequency-dependent behavior, which impacts their resistance, inductance, capacitance, and conductance. As the frequency of the signal increases, the skin effect becomes more pronounced, causing current to flow primarily at the surface of the conductor. This effect effectively reduces the usable cross-sectional area of the conductor, thereby increasing its resistance. At the frequency specified in the exercise, 400 MHz, the skin depth (\(\delta\)) becomes much smaller than the thickness of the conductors, ensuring that the skin effect is a dominant factor to consider. The skin effect can thus lead to higher losses in transmission lines as the frequency increases, necessitating careful design to mitigate these losses, such as choosing materials with a low resistivity like copper and ensuring the conductors are sufficiently thick to handle the skin effect at the operating frequency.