Examkin

Question

Difficulty: MediumResistance and Resistivity

A technician needs to replace a copper feeder cable with an aluminum cable of identical length while maintaining the exact same total electrical resistance. Since aluminum has a higher electrical resistivity than copper (ρAl>ρCu\rho_{\text{Al}} > \rho_{\text{Cu}}), which of the following physical characteristics must the replacement aluminum cable have?

  1. A
    A smaller cross-sectional area than the original copper cable
  2. A larger cross-sectional area than the original copper cableAnswer
  3. C
    A higher American Wire Gauge (AWG) number than the original copper cable
  4. D
    An applied potential difference inversely proportional to the material resistivity

Answer

A larger cross-sectional area than the original copper cable
The electrical resistance of a conductor is given by the formula R=ρLAR = \rho \frac{L}{A}. When replacing a material with one having a higher resistivity (ρ\rho), the cross-sectional area (AA) must be increased proportionally to maintain the same total resistance (RR) over an identical length (LL). Therefore, a thicker conductor with a larger cross-sectional area is required.

Step-by-Step Solution

1
Identify the mathematical relationship governing conductor resistance
R=ρLAR = \rho \frac{L}{A} where RR is resistance, ρ\rho is resistivity, LL is length, and AA is cross-sectional area
Electrical resistance depends directly on material resistivity and length, and inversely on cross-sectional area.
2
Analyze the constraints given in the problem
RAl=RCuR_{\text{Al}} = R_{\text{Cu}}, LAl=LCuL_{\text{Al}} = L_{\text{Cu}}, and ρAl>ρCu\rho_{\text{Al}} > \rho_{\text{Cu}}
The replacement wire must preserve equal total resistance and length despite the change in material.
3
Determine the required adjustment to cross-sectional area
AAl=ρAlρCu⋅ACu>ACuA_{\text{Al}} = \frac{\rho_{\text{Al}}}{\rho_{\text{Cu}}} \cdot A_{\text{Cu}} > A_{\text{Cu}}
To counterbalance the higher resistivity ρAl\rho_{\text{Al}}, the denominator AA must increase proportionally.

Key Concept

Conductor Resistance and Material Resistivity
Estimated Time:1m 0s
Rate this question

Topics

Resistance and Resistivity

More questions from this topic