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Difficulty: MediumResistance and Resistivity

An avionics technician replaces a copper ground wire in an aircraft instrument panel. The replacement copper wire is twice as long and has twice the diameter of the original wire. If the original wire had a measured resistance of 0.4 Ω0.4\ \Omega, what is the electrical resistance of the replacement wire?

  1. A
    0.8 Ω0.8\ \Omega
  2. 0.2 Ω0.2\ \OmegaAnswer
  3. C
    0.4 Ω0.4\ \Omega
  4. D
    1.6 Ω1.6\ \Omega

Answer

The electrical resistance of the replacement wire is 0.2 Ω0.2\ \Omega.
The electrical resistance of a conductor is determined by the formula R=ρLAR = \rho \frac{L}{A}, where ρ\rho is resistivity, LL is length, and AA is cross-sectional area. Because the wire cross-section is circular, A=πd24A = \frac{\pi d^2}{4}, meaning area scales with the square of diameter (d2d^2). Doubling the diameter increases the cross-sectional area by a factor of 22=42^2 = 4. Doubling the length increases resistance by a factor of 22. Combining these gives a new resistance of 24×0.4 Ω=0.5×0.4 Ω=0.2 Ω\frac{2}{4} \times 0.4\ \Omega = 0.5 \times 0.4\ \Omega = 0.2\ \Omega.

Step-by-Step Solution

1
Identify the relationship between conductor dimensions and resistance
The resistance formula is R=ρLAR = \rho \frac{L}{A}, where cross-sectional area is A=πd24A = \frac{\pi d^2}{4}.
Resistance varies directly with length LL and inversely with cross-sectional area AA.
2
Determine the change in cross-sectional area when diameter doubles
Since A∝d2A \propto d^2, doubling the diameter (2d2d) increases cross-sectional area by a factor of 22=42^2 = 4.
Area is proportional to the square of the diameter for a circular cross-section.
3
Calculate the scaling factor for total resistance
The new resistance scales by length factorarea factor=24=0.5\frac{\text{length factor}}{\text{area factor}} = \frac{2}{4} = 0.5.
Doubling length multiplies resistance by 22, and quadrupling area divides resistance by 44.
4
Calculate the final resistance of the replacement wire
Rnew=0.5×0.4 Ω=0.2 ΩR_{\text{new}} = 0.5 \times 0.4\ \Omega = 0.2\ \Omega.
Multiplying the baseline resistance by the net geometric scaling factor yields the new resistance value.

Key Concept

Conductor resistance is directly proportional to length and inversely proportional to the square of its diameter (R=ρL/AR = \rho L / A).
Estimated Time:1m 0s
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Topics

Resistance and Resistivity

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