Final answer:
The resistance of a copper wire of 23.8m length is 0.0589 Ω. The diameter of a copper wire that has the same resistance as an equal length of aluminum wire is 3.69mm.
Step-by-step explanation:
To find the resistance of a copper wire, we can use the formula:
R = (ρL) / A,
where R is the resistance, ρ is the resistivity of copper (1.68 x 10^-8 Ωm), L is the length of the wire (23.8 m), and A is the cross-sectional area of the wire.
The cross-sectional area of the wire can be calculated using the formula:
A = πr^2, where r is the radius of the wire.
Given that the diameter of the wire is 2.05mm, the radius can be calculated as
r = (2.05mm / 2) = 1.025mm = 0.001025m.
Substituting the values into the formulas, we get:
R = (1.68 x 10^-8 Ωm * 23.8m) / (π * (0.001025m)^2) = 0.0589 Ω
To find the diameter of a copper wire that has the same resistance as an equal length of aluminum wire with a diameter of 3.84mm, we can use the formula for the resistance:
R = (ρL) / A.
Since the resistance should be the same, we can set up the equation:
(1.68 x 10^-8 Ωm * L) / (π * (0.001025m)^2) = (2.82 x 10^-8 Ωm * L) / (π * (d/2)^2)
By rearranging the equation, we can solve for the diameter d:
d = √((1.68 x 10^-8 m * 3.84mm^2) / (2.82 x 10^-8 m)) = 3.69mm