If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?

If aluminum and copper wires of the same length have the same resistance, the aluminum wire must have a larger diameter than the copper wire. This is because the resistance of a wire is inversely proportional to its cross-sectional area, and since aluminum has a lower electrical conductivity than copper, it needs a larger cross-sectional area to achieve the same resistance.

Here’s the reasoning:

1. Resistance Formula:
   The resistance $R$ of a wire is given by the formula:

   R=ρ.L/A​

   where:

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

2. Resistivity:

   • The resistivity of copper is lower than that of aluminum. Copper’s resistivity is about 1.68×10−8 Ω⋅m, while aluminum’s resistivity is about 2.82×10−8 Ω⋅m2.

3. Cross-sectional Area:
   Since the resistances of the two wires are the same, we can set up the equation for both materials:

   ρ_CuL/A_Cu=ρ_AlL/A_Al​

   Canceling the length ($L$) on both sides:

   ρ_Cu/A_Cu=ρ_Al/A_Al​​

   This can be rewritten as:

   A_Al=A_Cu×ρ_Cuρ_Al​​

   Since ρ_Cu<ρ_Al, A_Al>A_Cu​.

4. Diameter Relationship:
   The cross-sectional area $A$ of a wire is related to its diameter $d$ by the formula A=πd²/4. Therefore, a larger area corresponds to a larger diameter, the diameter of the aluminum wire will be larger than that of the copper wire to maintain the same resistance.

In summary, the aluminum wire must have a larger diameter because its resistivity is higher, and a larger cross-sectional area (and therefore diameter) is needed to achieve the same resistance as the copper wire.