Capacitance per unit length of coaxial cable in vacuum is C' = (2π ε0)/ln(b/a). If the cable length is L, what is the total capacitance?

Capacitance per unit length of coaxial cable in vacuum is C' = (2π ε0)/ln(b/a). If the cable length is L, what is the total capacitance?

• A. C = (2π ε0 L)/ln(b/a)
• B. C = (2π ε0)/ (L ln(b/a))
• C. C = (ε0 L)/ln(b/a)
• D. C = (2π ε0 L) ln(b/a)

Answer: (A)

Capacitance per unit length tells you how much charge is stored per volt for each meter of the line. To get the total capacitance of a cable of length L, you multiply by the length because each additional meter adds the same amount of capacitance. So C = C' × L. Plugging in C' = (2π ε0)/ln(b/a) gives C = [(2π ε0)/ln(b/a)] × L = (2π ε0 L)/ln(b/a). The ln term and the radii a and b capture the coaxial geometry and stay the same regardless of length, while the length factor simply scales the capacitance linearly. This matches the idea that doubling the length doubles the total capacitance.

Capacitance per unit length tells you how much charge is stored per volt for each meter of the line. To get the total capacitance of a cable of length L, you multiply by the length because each additional meter adds the same amount of capacitance. So C = C' × L.

Plugging in C' = (2π ε0)/ln(b/a) gives C = [(2π ε0)/ln(b/a)] × L = (2π ε0 L)/ln(b/a). The ln term and the radii a and b capture the coaxial geometry and stay the same regardless of length, while the length factor simply scales the capacitance linearly. This matches the idea that doubling the length doubles the total capacitance.