Final answer:
To find the diameter of a copper wire that has the same resistance as an equal length of aluminum wire with a given diameter, we can use the formula for resistance of a wire and rearrange the equation to solve for the diameter of the copper wire.
Step-by-step explanation:
To find the diameter of a copper wire that has the same resistance as an equal length of aluminum wire with a given diameter, we can use the formula for the resistance of a wire, which is R = (ρL)/(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. Since we want the resistance to be the same for both wires, we can set up the equation (ρCopper)(LCopper)/(DCopper^2) = (ρAluminum)(LAluminum)/(DAluminum^2), where DCopper is the diameter of the copper wire and DAluminum is the diameter of the aluminum wire.
We can rearrange the equation to solve for the diameter of the copper wire: DCopper^2 = (ρCopper/ρAluminum) * (LAluminum/LCopper) * DAluminum^2. Plugging in the given values, including the diameter of the aluminum wire, we can solve for the diameter of the copper wire.
Let's substitute the given values:
ρCopper = resistivity of copper
ρAluminum = resistivity of aluminum
LCopper = length of copper wire
LAluminum = length of aluminum wire
DAluminum = diameter of aluminum wire
After substituting the values and calculating, we can find the diameter of the copper wire that has the same resistance as the given diameter of the aluminum wire.