Answer:
The resistance (R) of 1 mile of copper wire is approximately 83.05 ohms, and the current density (J) in the wire is approximately 3.034x10^7 A/m^2.
Step-by-step explanation:
To find the resistance (R) and current density (J) of the copper wire, we can follow these steps:
Step 1: Calculate the cross-sectional area of the copper wire.
Step 2: Find the resistance of 1 mile of copper wire.
Step 3: Calculate the current density.
Step 1: Calculate the cross-sectional area (A) of the copper wire:
The diameter of the copper wire (d) is given as 1.291×10^-3 m. The radius (r) is half of the diameter:
r = d / 2 = 1.291×10^-3 m / 2 = 6.455×10^-4 m.
The cross-sectional area (A) of the wire is given by the formula for the area of a circle:
A = π * r^2 = 3.14159 * (6.455×10^-4 m)^2 ≈ 3.298×10^-7 m^2.
Step 2: Find the resistance (R) of 1 mile of copper wire:
The resistivity of copper (ρ) is approximately 1.7x10^-8 ohm-meter.
The resistance (R) of the copper wire can be calculated using the formula:
R = (ρ * L) / A,
where L is the length of the wire and is given as 1 mile, which is 1609 meters.
R = (1.7x10^-8 ohm-meter * 1609 m) / 3.298×10^-7 m^2
R ≈ 83.05 ohms.
Step 3: Calculate the current density (J):
Current density (J) is defined as the current passing through a unit area perpendicular to the direction of current flow. It is given by the formula:
J = I / A,
where I is the current flowing through the wire (given as 10A), and A is the cross-sectional area we calculated earlier.
J = 10A / 3.298×10^-7 m^2 ≈ 3.034x10^7 A/m^2.
So, the resistance (R) of 1 mile of copper wire is approximately 83.05 ohms, and the current density (J) in the wire is approximately 3.034x10^7 A/m^2.