Understanding the Ratio of Resistance in Copper Wires of Different Thicknesses

When working with wires, particularly those made of copper, it's important to understand how the cross-sectional area affects resistance. This article explains the fundamental relationship between the cross-sectional area and resistance, along with a practical example calculation.

Theoretical Background

The resistance (R) of a conducting wire is a function of the material's resistivity (ρ), the wire's length (L), and its cross-sectional area (A). The relationship is given by:

R ρL / A

Key Parameters and Definitions

R (Resistance): The opposition to the flow of electric current. ρ (Resistivity): The inherent property of a material that determines its resistance to electric current. L (Length): The physical length of the wire. A (Cross-Sectional Area): The area of the wire perpendicular to the direction of the flow of current.

For copper, the resistivity (ρ) is approximately 1.68 × 10-8 Omega;·m.

Calculation Example: 1 mm and 2 mm Thick Copper Wires

Let's consider two copper wires of different diameters: one with a diameter of 1 mm and another with a diameter of 2 mm. Both wires will be of the same length, which does not affect the resistance ratio calculation.

Step 1: Calculate the Cross-Sectional Areas

The cross-sectional area (A) of a wire is calculated using the formula:

A (πd2) / 4

where d is the diameter of the wire.

For the 1 mm wire:

Diameter (d1) 1 mm 0.001 m Area (A1) (π(0.001)2) / 4 (π × 10-6) / 4 ≈ 7.85 × 10-7 m2

For the 2 mm wire:

Diameter (d2) 2 mm 0.002 m Area (A2) (π(0.002)2) / 4 (π × 4 × 10-6) / 4 π × 10-6 ≈ 3.14 × 10-6 m2

Step 2: Determine the Ratio of Resistances

Since both wires are made of the same material and of the same length, the ratio of their resistances is the inverse of the ratio of their cross-sectional areas:

frac{R1}{R2} frac{A2}{A1}

Substituting the areas we calculated:

frac{R1}{R2} frac{3.14 × 10-6}{7.85 × 10-7} ≈ 4

Therefore, the ratio of resistance for the two wires is approximately 1 : 4. This means that the 1 mm wire has four times the resistance of the 2 mm wire.

Conclusion

In summary, the ratio of resistance between two copper wires of different thickness (diameters) is determined by the ratio of their cross-sectional areas. A 1 mm diameter wire will have roughly four times the resistance of a 2 mm diameter wire, assuming both are of the same length and made of the same material.