A solid copper wire used in an industrial application has a length of 200\text{ meters} an… — Armed Services Vocational Aptitude Battery (ASVAB)

A solid copper wire used in an industrial application has a length of $200\text{ meters}$ and a cross-sectional area of $0.85 \, mm^2$ . The resistivity of copper at $20^\circ C$ is $1.7 \times 10^{-8} \, \Omega\cdot m$ , and its temperature coefficient of resistance is $0.004 \text{ per } ^\circ C$ . What is the resistance of this wire, in ohms, when its operating temperature reaches $70^\circ C$ ?

Cevap: 4.8 \Omega

Cevap

4.8

The solution requires a synthesis of two core resistance principles. First, the baseline resistance is found by dividing the length by the area (converted to square meters) and multiplying by the resistivity, yielding 4.0 ohms. Second, the 50°C temperature increase is multiplied by the coefficient (0.004) to find a 20% increase in total resistance, raising the final value to 4.8 ohms.

Adım Adım Çözüm

Convert the cross-sectional area from

mm^2

m^2

0.85 \, mm^2 = 0.85 \times 10^{-6} \, m^2

Standard SI units must be used to cancel with the meters in the resistivity value.

Calculate the resistance at

20^\circ C

(

R_{20}

) using Pouillet's law (

R = \rho \frac{L}{A}

R_{20} = (1.7 \times 10^{-8}) \times \frac{200}{0.85 \times 10^{-6}} = 4.0 \, \Omega

This establishes the baseline resistance of the wire before temperature effects are considered.

Calculate the change in temperature (

\Delta T

\Delta T = 70^\circ C - 20^\circ C = 50^\circ C

The temperature coefficient is applied to the difference between the operating temperature and the reference temperature.

Calculate the new resistance at

70^\circ C

using the temperature coefficient formula:

R_{new} = R_{ref}[1 + \alpha(\Delta T)]

R_{70} = 4.0 \times [1 + 0.004 \times 50] = 4.0 \times [1 + 0.2] = 4.0 \times 1.2 = 4.8 \, \Omega

As temperature increases, the resistance of a standard conductor like copper increases proportionally.

Anahtar Kavram

Calculating absolute resistance using dimensions and resistivity, followed by applying temperature coefficients.

Tahmini Süre:3m 0s

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