Question

What diameter must a copper wire have if its resistance is to be the same as that of an equal length of aluminum wire with diameter 1.74 mm ? Express your answer with the appropriate units.

Answer 1

To find the diameter of the copper wire that has the same resistance as an equal length of aluminum wire, we can use the formula for resistance and the fact that their resistivities and lengths are equal. The resistance is directly proportional to the length and inversely proportional to the cross-sectional area. By setting up an equation using the resistivity and cross-sectional areas of the copper and aluminum wires, we can solve for the diameter of the copper wire.

Step-by-step explanation:

To find the diameter of the copper wire, we can use the fact that the resistance of two wires is the same if their lengths and resistivities are equal. The resistance of a wire is directly proportional to its length and inversely proportional to its cross-sectional area. The formula for resistance is R = ρL/A, where R is the resistance, ρ is the resistivity, L is the length, and A is the cross-sectional area.

Let's assume the resistivity of copper is ρcopper and the resistivity of aluminum is ρaluminum. Since the resistance per unit length (R/L) is the same for both wires, we can set up the equation as follows: ρcopper / Acopper = ρaluminum / Aaluminum.

We know the diameter of the aluminum wire (1.74 mm), so we can calculate its cross-sectional area using the formula Aaluminum = π(Daluminum/2)2. Solving for Acopper using the given equation, we can then find the diameter (Dcopper) of the copper wire using the formula Dcopper = 2 √(Acopper/π).

Let's substitute the values into the equations:

1. ρaluminum = resistivity of aluminum = value in ohm-m
2. dialuminium = diameter of aluminum wire = 1.74 mm = value in meters
3. ρcopper = resistivity of copper = value in ohm-m

After substituting the values, you can calculate the diameter of the copper wire.