Researchers study the electrical resistance of four different wire samples as a function o… — ACT

Researchers study the electrical resistance of four different wire samples as a function of temperature. The resistance $R$ (in ohms, $\Omega$ ) of a metal wire at temperature $T$ (in $^\circ\text{C}$ ) is modeled by the linear relationship:

R(T) = R_0 [1 + \alpha(T - T_0)]

where $R_0$ is the baseline resistance at the reference temperature $T_0 = 20^\circ\text{C}$ , and $\alpha$ is the temperature coefficient of resistance (in $^\circ\text{C}^{-1}$ ).

The baseline resistance and temperature coefficients for the four wire samples are shown in the table below:

Wire Material	Baseline Resistance ( $R_0$ at $20^\circ\text{C}$ )	Temperature Coefficient ( $\alpha$ )
Copper	$10\ \Omega$	$0.0040\ ^\circ\text{C}^{-1}$
Iron	$8\ \Omega$	$0.0060\ ^\circ\text{C}^{-1}$
Tungsten	$12\ \Omega$	$0.0045\ ^\circ\text{C}^{-1}$
Carbon	$15\ \Omega$	$-0.0005\ ^\circ\text{C}^{-1}$

Arrange the wire samples in order of their predicted electrical resistance at $120^\circ\text{C}$ from lowest resistance to highest resistance.

1Iron wire
2Copper wire
3Carbon wire
4Tungsten wire

Cevap

The correct order of wire samples from lowest to highest predicted resistance is Iron wire, Copper wire, Carbon wire, and Tungsten wire.

Evaluating the linear resistance model at

120^\circ\text{C}

(

T - T_0 = 100^\circ\text{C}

) yields resistances of

12.8\ \Omega

for Iron,

14.0\ \Omega

for Copper,

14.25\ \Omega

for Carbon, and

17.4\ \Omega

for Tungsten, establishing the sequence from lowest to highest.

Adım Adım Çözüm

Calculate the temperature difference (

T - T_0

) from the reference temperature to the target temperature.

T - T_0 = 120^\circ\text{C} - 20^\circ\text{C} = 100^\circ\text{C}

The model uses the temperature deviation from the baseline reference temperature

T_0 = 20^\circ\text{C}

Apply the model

R(T) = R_0 [1 + \alpha(T - T_0)]

to calculate the resistance of each wire at

120^\circ\text{C}

Iron:

8 \times [1 + 0.0060(100)] = 12.8\ \Omega

; Copper:

10 \times [1 + 0.0040(100)] = 14.0\ \Omega

; Carbon:

15 \times [1 - 0.0005(100)] = 14.25\ \Omega

; Tungsten:

12 \times [1 + 0.0045(100)] = 17.4\ \Omega

Evaluating the mathematical model with the given parameters determines the resistance values for comparison.

Order the wire materials from the smallest calculated resistance value to the largest.

Iron (

12.8\ \Omega

) < Copper (

14.0\ \Omega

) < Carbon (

14.25\ \Omega

) < Tungsten (

17.4\ \Omega

This matches the requested sorting direction from lowest to highest resistance.

Anahtar Kavram

Applying mathematical linear models to predict physical quantities at specific conditions.

Tahmini Süre:1m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru