Calculating DC Resistance of Copper Wire

Understanding the resistance of a conductor like a copper wire is fundamental in electrical engineering and physics. The resistance of a material depends on its physical dimensions and a property called resistivity. For a uniform conductor, the resistance can be calculated using a specific formula.

Understanding Electrical Resistance and Resistivity

Electrical resistance (\(R\)) is a measure of how much a material opposes the flow of electric current. Resistivity (\(\rho\)) is an intrinsic property of the material itself, indicating its resistance per unit length and cross-sectional area. The relationship is given by the formula:

\[ R = \rho \frac{L}{A} \]

Where:

• \(R\) is the resistance in Ohms (\(\Omega\)).
• \(\rho\) is the resistivity of the material in Ohm-metre (\(\Omega \cdot \text{m}\)).
• \(L\) is the length of the conductor in metres (\(\text{m}\)).
• \(A\) is the cross-sectional area of the conductor in square metres (\(\text{m}^2\)).

Given Parameters

From the question, we are given the following information for the copper wire:

• Length (\(L\)) = 100 metre
• Cross-sectional Area (\(A\)) = 2.5 mm²
• Resistivity of copper (\(\rho\)) at 20° C = 1.72 × 10^-8 \(\Omega \cdot \text{m}\)

Unit Conversion

Before using the formula, we must ensure all units are consistent. The resistivity is given in \(\Omega \cdot \text{m}\), and the length is in metres, but the area is in mm². We need to convert the cross-sectional area from mm² to m².

We know that 1 metre = 1000 millimetres.

Therefore, 1 m² = (1000 mm)² = 1,000,000 mm² = 10⁶ mm².

To convert from mm² to m², we divide by 10⁶.

So, \(A = 2.5 \text{ mm}^2 = 2.5 \times \frac{1}{10^6} \text{ m}^2 = 2.5 \times 10^{-6} \text{ m}^2\).

Step-by-Step Calculation of DC Resistance

Now we can substitute the values into the resistance formula \(R = \rho \frac{L}{A}\):

\[ R = (1.72 \times 10^{-8} \, \Omega \cdot \text{m}) \times \frac{100 \, \text{m}}{2.5 \times 10^{-6} \, \text{m}^2} \]

\[ R = (1.72 \times 10^{-8}) \times \frac{100}{2.5 \times 10^{-6}} \, \Omega \]

Let's simplify the fraction part first:

\[ \frac{100}{2.5 \times 10^{-6}} = \frac{10^2}{2.5 \times 10^{-6}} = \frac{10^2}{2.5} \times \frac{1}{10^{-6}} = \frac{100}{2.5} \times 10^6 = 40 \times 10^6 \]

Now substitute this back into the resistance formula:

\[ R = (1.72 \times 10^{-8}) \times (40 \times 10^6) \, \Omega \]

\[ R = (1.72 \times 40) \times (10^{-8} \times 10^6) \, \Omega \]

\[ R = 68.8 \times 10^{-8+6} \, \Omega \]

\[ R = 68.8 \times 10^{-2} \, \Omega \]

\[ R = 0.688 \, \Omega \]

Conclusion

The calculated total DC resistance of the 100-metre roll of 2.5 mm² copper wire is 0.688 \(\Omega\).

Revision Table: Copper Wire Resistance Calculation

Additional Information: Factors Affecting Resistance

Besides resistivity, length, and cross-sectional area, the resistance of a conductor can also be affected by temperature. For most conductors, including copper, resistance increases with increasing temperature. The resistivity value provided in the question (at 20° C) is a standard reference point. The resistance at a different temperature can be estimated using the temperature coefficient of resistivity for the material.

Understanding these factors is crucial when working with electrical circuits, especially for long wires or in environments with varying temperatures.