The characteristic impedance of a co-axial cable is between (A) \(10 \Omega\) to \(20 \Omega\) (B) \(20 \Omega\) to \(30 \Omega\) (C) \(30 \Omega\) to \(60 \Omega\) (D) \(50 \Omega\) to \(70 \Omega\)

The characteristic impedance (Z0) is an important value in transmission line theory, which is used to determine the behavior of a transmission line when it's connected to a load. It is also used when designing impedance matching circuits for maximum power transfer. For a coaxial cable, the characteristic impedance is defined as follows: \(Z_0 = \frac{1}{2 \pi} \sqrt{\frac{L}{C}}\) Here, L is the inductance per unit length and C is the capacitance per unit length of the coaxial cable.

Coaxial cables are used in a wide variety of applications, and their characteristic impedance can vary widely. However, there are some common standard industry values that most coaxial cables fall into. These include values such as 50 Ω, 60 Ω, 75 Ω, and even 93 Ω. The characteristic impedance of other types of transmission lines may be different. Now let's analyze the given options and see which range is correct: (A) 10 Ω to 20 Ω - Coaxial cables with such low impedances are very rare and not suitable for most industrial applications. Therefore, this range is unlikely. (B) 20 Ω to 30 Ω - This range also falls on the lower end of characteristic impedance values and is not commonly used in the industry. (C) 30 Ω to 60 Ω - This range includes the common industrial values of 50 Ω and 60 Ω. Many applications in telecommunications and digital networks use cables with an impedance of around 50 Ω. (D) 50 Ω to 70 Ω - This range also includes the common industrial value of 50 Ω, but cables with an impedance value of 70 Ω are less common. Based on our analysis, options (C) 30 Ω to 60 Ω, and (D) 50 Ω to 70 Ω are both considered valid ranges for the characteristic impedance of a coaxial cable. However, conflicting options should not have existed in a correctly formed multiple choice question. For this specific question, we will choose option (C) 30 Ω to 60 Ω, as it includes a wider range of common industry values. So, the correct answer is:

(C) \(30 \Omega\) to \(60 \Omega\)