A transmission line cable consists of 26 strands of identical copper conductors, each 1.255 mm in diameter. The length of the cable is 10 km, but because of the twist of the strands, the actual length of each conductor is increased by 5%. If the resistivity of the copper is 1.72 x 10-8 Ω.m, what is the resistance of the cable in ohms?

To solve this problem, we need to use the formula for resistance, which is:

R = ρ * (L/A)

where:

• R is the resistance,
• ρ is the resistivity,
• L is the length of the conductor, and
• A is the cross-sectional area of the conductor.

Step 1: Calculate the cross-sectional area of one strand. The cross-sectional area (A) of a circular conductor is given by the formula A = π * (d/2)^2, where d is the diameter of the conductor. Substituting the given diameter:

A = π * (1.255 x 10^-3 m / 2)^2 = 1.236 x 10^-6 m^2

Step 2: Calculate the total cross-sectional area. Since there are 26 strands, the total cross-sectional area is 26 * A = 26 * 1.236 x 10^-6 m^2 = 3.2136 x 10^-5 m^2

Step 3: Calculate the length of the conductor. The actual length of each conductor is increased by 5%, so the length of one conductor is 10 km * 1.05 = 10.5 km = 10.5 x 10^3 m.

Step 4: Calculate the resistance. Substitute the values into the resistance formula:

R = ρ * (L/A) = 1.72 x 10^-8 Ω.m * (10.5 x 10^3 m / 3.2136 x 10^-5 m^2) = 5.63 Ω

So, the resistance of the cable is approximately 5.63 ohms.

To solve this problem, we need to follow these steps:

1. Calculate the cross-sectional area of one strand: The formula for the area of a circle is A = πr². The radius is half the diameter, so r = 1.255 mm / 2 = 0.6275 mm = 0.0006275 m. Therefore, A = π * (0.0006275 m)² = 1.237 x 10^-6 m².

2. Calculate the total cross-sectional area of all strands: Since there are 26 strands, the total area is 26 * 1.237 x 10^-6 m² = 3.2162 x 10^-5 m².

3. Calculate the length of one strand: The length of the cable is increased by 5% due to the twist of the strands, so the actual length is 10 km * 1.05 = 10.5 km = 10500 m.

4. Calculate the resistance of the cable: The formula for resistance is R = ρL/A, where ρ is resistivity, L is length, and A is area. Substituting the given values, we get R = 1.72 x 10^-8 Ω.m * 10500 m / 3.2162 x 10^-5 m² = 0.056 Ω.

So, the resistance of the cable is 0.056 ohms.