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Zorluk: OrtaResistance and Resistivity

A 10-meter copper conductor has a measured resistance of 0.05 Ω0.05\ \Omega. A technician needs to fabricate a replacement wire of the exact same length and resistance using an alloy with 4 times the resistivity of copper. How must the cross-sectional area of the replacement wire compare to the original copper wire?

  1. A
    The cross-sectional area must be reduced to 1/4 of the original wire.
  2. B
    The cross-sectional area must remain exactly the same as the original wire.
  3. The cross-sectional area must be 4 times larger than the original wire.Cevap
  4. D
    The cross-sectional area must be 16 times larger than the original wire.

Cevap

The cross-sectional area must be 4 times larger than the original wire.
Based on the resistance formula R=ρLAR = \rho \frac{L}{A}, resistance is directly proportional to resistivity ρ\rho and inversely proportional to cross-sectional area AA. When the replacement wire material has 4 times the resistivity (ρ′=4ρ\rho' = 4\rho) but must maintain identical length LL and total resistance RR, the cross-sectional area must increase by a factor of 4 (A′=4AA' = 4A) to counterbalance the higher material resistivity.

Adım Adım Çözüm

1
Identify the relationship between electrical resistance, material resistivity, length, and cross-sectional area.
R=ρLAR = \rho \frac{L}{A}, where RR is resistance, ρ\rho is resistivity, LL is wire length, and AA is cross-sectional area.
This formula defines how physical geometry and material properties determine electrical resistance.
2
Rearrange the equation to express cross-sectional area as a function of resistivity, length, and resistance.
A=ρLRA = \frac{\rho L}{R}
Isolating cross-sectional area enables direct evaluation of how area must adjust when resistivity changes while length and resistance remain constant.
3
Substitute the scaled resistivity ρ′=4ρ\rho' = 4\rho into the rearranged formula with constant length LL and constant resistance RR.
A′=(4ρ)LR=4(ρLR)=4AA' = \frac{(4\rho)L}{R} = 4\left(\frac{\rho L}{R}\right) = 4A
Because cross-sectional area is directly proportional to resistivity under constant length and resistance, quadrupling resistivity requires quadrupling the area.

Anahtar Kavram

Resistance and Resistivity Proportionality
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Resistance and Resistivity

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