A hollow tubular conductor is constructed from a type of brass having a conductivity of \(1.2 \times 10^{7} \mathrm{~S} / \mathrm{m} .\) The inner and outer radii are 9 and \(10 \mathrm{~mm}\), respectively. Calculate the resistance per meter length at a frequency of (a) \(\mathrm{dc} ;\) (b) \(20 \mathrm{MHz}\); (c) \(2 \mathrm{GHz}\).

Understanding the basics of conductivity and resistivity is crucial in the field of electrical engineering. Conductivity, denoted by the symbol \(\sigma\), indicates how easily a material can allow the flow of electric current. It is measured in siemens per meter (S/m). The higher the conductivity, the less resistance the material presents to the flow of electric charge.

On the flip side, resistivity, represented by the symbol \(\rho\), is the measure of a material's tendency to resist the flow of electric current. It is the inverse of conductivity, with the unit of ohm-meters (\(\Omega\cdot m\)). Resistivity explains why materials like copper and aluminum are commonly used in electrical wiring—their low resistivity means they allow electric current to flow easily.

To calculate the resistance of a conductor, you can use the formula \(R = \rho \cdot (\frac{L}{A})\), where \(L\) is the length of the conductor, and \(A\) is the cross-sectional area. For the hollow tubular conductor made of brass, the given conductivity value is \(1.2 \times 10^{7} S/m\), making the resistivity \(\frac{1}{1.2 \times 10^{7}} \approx 8.33 \times 10^{-8} \Omega m\).

In practical terms, selecting a conductor for a particular application involves striking a balance between cost, conductivity, and other physical properties.

One important phenomenon to consider in high-frequency electrical circuits is the skin effect. At low frequencies, the distribution of current throughout a conductor is relatively uniform. However, as the frequency increases, the current tends to be concentrated near the surface of the conductor, effectively reducing the area through which the current flows. The reason for this behavior lies in the electromagnetic fields generated by the alternating current, which induce eddy currents within the conductor and cause the current density to become greatest at the surface.

The depth at which the current density falls to \(1/e\) (about 37% of its surface value) is known as the skin depth, denoted by \(\delta\). It can be calculated using the formula \(\delta = \sqrt{\frac{2\rho}{\mu\omega}}\), where \(\omega\) is the angular frequency \(\omega = 2\pi f\), and \(\mu\) is the material's permeability.

For the hollow tubular conductor in the exercise, we observe a significant increase in resistance per meter as we move from DC to 20 MHz and then to 2 GHz, illustrating the pronounced effects of the skin effect at these higher frequencies. Since brass is non-magnetic, we use the permeability of free space (\(\mu_0 = 4\pi\times10^{-7} H/m\)) for calculations.

The frequency of the signal used in an electrical circuit has a direct influence on the resistance of conductors because of the skin effect. For direct current (DC) or very low frequencies, the skin effect is negligible, and the entire cross-section of the conductor participates in conducting current. But as the frequency increases, current concentrates near the surface, and the effective cross-sectional area of the conductor decreases.

This behavior has practical implications. For example, for applications with high-frequency signals (like the 20 MHz and 2 GHz signals in our exercise), engineers would need to consider the significant increase in resistance when designing circuits. The resistance doesn't scale linearly with frequency due to the complex interaction between the current and the conductor's magnetic fields.

As calculated in the exercise, the resistance per meter at 20 MHz is much higher than the resistance for DC. It further increases at 2 GHz. Precise understanding and management of resistance in relation to frequency are essential in radio frequency (RF) design, telecommunications, and high-speed digital circuits to ensure efficiency and reduce heat and power loss.