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Difficulty: EasyElectric Current and Resistance

A uniform conductor of length 50 m50\,\text{m} and cross-sectional area 2.0×10−6 m22.0 \times 10^{-6}\,\text{m}^2 is made of a material with a resistivity of 4.0×10−7 Ω⋅m4.0 \times 10^{-7}\,\Omega\cdot\text{m}. What is the electrical resistance of the conductor in ohms?

Answer: 10 ohms

Answer

The resistance of the conductor is 10 Ω10\,\Omega.
The resistance of a uniform conductor is given by R=ρLAR = \frac{\rho L}{A}. Substituting the values L=50 mL = 50\,\text{m}, A=2.0×10−6 m2A = 2.0 \times 10^{-6}\,\text{m}^2, and ρ=4.0×10−7 Ω⋅m\rho = 4.0 \times 10^{-7}\,\Omega\cdot\text{m} yields R=(4.0×10−7)(50)2.0×10−6=10 ΩR = \frac{(4.0 \times 10^{-7})(50)}{2.0 \times 10^{-6}} = 10\,\Omega.

Step-by-Step Solution

1
Identify given physical quantities
L=50 mL = 50\,\text{m}, A=2.0×10−6 m2A = 2.0 \times 10^{-6}\,\text{m}^2, ρ=4.0×10−7 Ω⋅m\rho = 4.0 \times 10^{-7}\,\Omega\cdot\text{m}
Extracting given parameter values clearly sets up the mathematical relationship.
2
Apply the resistivity formula for resistance
R=ρLAR = \frac{\rho L}{A}
Resistance varies directly with length and resistivity, and inversely with cross-sectional area.
3
Perform the calculation
R=(4.0×10−7 Ω⋅m)(50 m)2.0×10−6 m2=10 ΩR = \frac{(4.0 \times 10^{-7}\,\Omega\cdot\text{m})(50\,\text{m})}{2.0 \times 10^{-6}\,\text{m}^2} = 10\,\Omega
Multiplying the numerator gives 2.0×10−5 Ω⋅m22.0 \times 10^{-5}\,\Omega\cdot\text{m}^2; dividing by 2.0×10−6 m22.0 \times 10^{-6}\,\text{m}^2 yields 10 Ω10\,\Omega.

Key Concept

Direct computation of electrical resistance using resistivity, length, and cross-sectional area
Estimated Time:45s
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Topics

Electric Current and Resistance

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