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Difficulty: MediumElectric Current and Resistance

A uniform cylindrical wire of length 2.0 m2.0\,\text{m} and radius 1.0 mm1.0\,\text{mm} is connected to a direct-current source. When a potential difference of 12 V12\,\text{V} is applied across its ends, a steady current of 4.0 A4.0\,\text{A} flows through it. Taking π≈3.142\pi \approx 3.142, what is the resistivity of the material of the wire?

  1. 4.71×10−6 Ω⋅m4.71 \times 10^{-6}\,\Omega\cdot\text{m}Answer
  2. B
    4.71×10−3 Ω⋅m4.71 \times 10^{-3}\,\Omega\cdot\text{m}
  3. C
    1.89×10−5 Ω⋅m1.89 \times 10^{-5}\,\Omega\cdot\text{m}
  4. D
    9.43×10−6 Ω⋅m9.43 \times 10^{-6}\,\Omega\cdot\text{m}

Answer

The resistivity of the material of the wire is 4.71×10−6 Ω⋅m4.71 \times 10^{-6}\,\Omega\cdot\text{m}.
Using Ohm's law (R=V/IR = V/I), the wire's resistance is 3.0 Ω3.0\,\Omega. Substituting this resistance, the wire length (2.0 m2.0\,\text{m}), and the circular area (A=πr2=3.142×10−6 m2A = \pi r^2 = 3.142 \times 10^{-6}\,\text{m}^2) into ρ=RA/L\rho = R A / L yields 4.71×10−6 Ω⋅m4.71 \times 10^{-6}\,\Omega\cdot\text{m}.

Step-by-Step Solution

1
Calculate the resistance of the wire using Ohm's Law.
R=VI=12 V4.0 A=3.0 ΩR = \frac{V}{I} = \frac{12\,\text{V}}{4.0\,\text{A}} = 3.0\,\Omega
Ohm's Law relates potential difference, current, and resistance.
2
Calculate the cross-sectional area of the wire from its radius.
A=πr2=3.142×(1.0×10−3 m)2=3.142×10−6 m2A = \pi r^2 = 3.142 \times (1.0 \times 10^{-3}\,\text{m})^2 = 3.142 \times 10^{-6}\,\text{m}^2
The wire has a circular cross-section.
3
Rearrange the resistance formula R=ρLAR = \frac{\rho L}{A} to solve for resistivity ρ\rho.
ρ=RAL=3.0 Ω×3.142×10−6 m22.0 m=4.713×10−6 Ω⋅m≈4.71×10−6 Ω⋅m\rho = \frac{R A}{L} = \frac{3.0\,\Omega \times 3.142 \times 10^{-6}\,\text{m}^2}{2.0\,\text{m}} = 4.713 \times 10^{-6}\,\Omega\cdot\text{m} \approx 4.71 \times 10^{-6}\,\Omega\cdot\text{m}
Resistivity is an intrinsic property derived from resistance, length, and cross-sectional area.

Key Concept

Relationship between resistance, potential difference, current, and material resistivity.
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Topics

Electric Current and Resistance

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