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what length of copper wire, 0.462 mm in diameter, has a resistance of 1.00 ω?

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Question

what length of copper wire, 0.462 mm in diameter, has a resistance of 1.00 ω?

Answer:

The task that we have at hand involves determining the length of a copper wire with a diameter of 0.462 mm that exhibits a resistance of 1.00 Ω. Therefore, the length of the copper wire can be ascertained by employing the formula that relates resistance, length, and the material’s resistivity. The given resistance, R = 1.00 Ω, and the diameter, d = 0.462 mm. Thus, the final answer to the above question is 9.98 m.

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what length of copper wire, 0.462 mm in diameter, has a resistance of 1.00 ω?
what length of copper wire, 0.462 mm in diameter, has a resistance of 1.00 ω?

Text Explanation:

Data: Information revealed from the question

  • Resistance, R = 1.00 Ω
  • Diameter, d = 0.462 mm

Step 1: Convert Diameter to Radius
The first step involves converting the diameter to the radius (r). Given that the diameter (d) = 0.462 mm, the radius (r) = d/2 = 0.462/2 = 0.231 mm.

Step 2: Convert Radius to Meters
For consistency in units, we will need to convert the radius which is in millimetres (mm) into meters (m) by dividing it by 1000. This is because 1 m equals 1000 mm.

Therefore, r = 0.231 mm / 1000 = 0.000231 m.

Step 3: Calculate Cross-Sectional Area
In addition to what we have done, we can now determine the cross-sectional area (A) of the wire by using the formula for the area of a circle, where A = πr2.

Step 4: Apply Ohm’s Law
In this step, we will apply the formula for resistivity, which is  ρ = RA/L

We will need to rearrange the above formula to solve for length (L). Thus, L = RA/ρ

However, ρ is the resistivity of copper, a constant (1.68 x 10-8 Ωm)

Solution:

Step 5: Substitute Values From Data and Solve for Length
L = RA/ρ = (πr2 x 1Ω) / (1.68 x 10-8 Ωm) [Remember that A = πr2]

Additionally, since r = 0.000231 m = 2.31 x 10-4 m

and π = 3.142

We can comfortably write:

L = = (πr2 x 1Ω) / (1.68 x 10-8 Ωm) = (3.142 x (0.000231m)2 x 1Ω) / (1.68 x 10-8 Ωm)

Therefore, L = 9.98 m

Thus, upon solving for Length ( L ), the length of the copper wire is found to be approximately 9.98 meters.

Therefore, a copper wire with a diameter of 0.462mm and a resistance of 1.00Ω has a length of approximately 9.98m.

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