In case of overhead lines for the same resistance, the diameter of aluminium conductor is about ______ times the diameter of copper conductor.  

Explanation:

Diameter Comparison between Aluminium and Copper Conductors

Definition: In electrical engineering, the diameter of a conductor plays a crucial role in determining its resistance, current-carrying capacity, and mechanical strength. Aluminium and copper are two commonly used materials for conductors in overhead lines. Although aluminium has a lower electrical conductivity compared to copper, it is often used due to its lower cost and lighter weight. When comparing the diameters of aluminium and copper conductors that have the same resistance, certain principles and calculations must be considered.

Resistance and Material Properties:

The resistance (R) of a conductor is given by the formula:

R = ρ (L / A)

where:

• ρ (rho) is the resistivity of the material (Ω·m)
• L is the length of the conductor (m)
• A is the cross-sectional area of the conductor (m²)

For the same resistance and length, the cross-sectional area is inversely proportional to the resistivity. This relationship can be expressed as:

A_al / A_cu = ρ_al / ρ_cu

where:

• A_al and A_cu are the cross-sectional areas of the aluminium and copper conductors, respectively
• ρ_al and ρ_cu are the resistivities of aluminium and copper, respectively

Resistivity Values:

• Resistivity of copper (ρ_cu): approximately 1.68 × 10^-8 Ω·m
• Resistivity of aluminium (ρ_al): approximately 2.82 × 10^-8 Ω·m

Using the ratio of resistivities:

A_al / A_cu = 2.82 × 10^-8 / 1.68 × 10^-8 ≈ 1.68

Cross-Sectional Area and Diameter:

The cross-sectional area of a conductor with a circular cross-section is given by:

A = π (d / 2)²

where:

• d is the diameter of the conductor

If the areas are proportional to the diameters squared, then:

(d_al / d_cu)² = A_al / A_cu ≈ 1.68

Taking the square root of both sides:

d_al / d_cu = √1.68 ≈ 1.3

Thus, the diameter of the aluminium conductor is approximately 1.3 times the diameter of the copper conductor for the same resistance.

Correct Option Analysis:

The correct option is:

Option 3: 1.26

This option is the closest to the calculated value of 1.3, considering practical rounding and slight variations in resistivity values used in different sources. Therefore, the diameter of the aluminium conductor is approximately 1.26 times the diameter of the copper conductor for the same resistance.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 1: 4.26

This option is incorrect. The ratio of diameters based on the resistivity values does not support such a high factor. The calculation clearly shows that the diameter ratio is much closer to 1.3, making 4.26 an unrealistic option.

Option 2: 3.26

Similar to option 1, this value is significantly higher than the calculated ratio. The relationship between resistivity and cross-sectional area does not justify a diameter ratio of 3.26.

Option 4: 2.26

This value is also too high. While it is closer than options 1 and 2, it still exceeds the calculated diameter ratio of approximately 1.3.

Conclusion:

Understanding the relationship between the resistivity, cross-sectional area, and diameter of conductors is essential for determining the appropriate size of conductors in electrical applications. For aluminium and copper conductors with the same resistance, the diameter of the aluminium conductor is approximately 1.26 times the diameter of the copper conductor. This insight is valuable for designing efficient and cost-effective electrical systems, ensuring that the chosen materials meet the required electrical and mechanical specifications.