Examkin

Question

Difficulty: MediumElectric Current and Resistance

A resistance thermometer has a resistance of 5.0 Ω5.0\,\Omega at 0∘C0^\circ\text{C} and 5.8 Ω5.8\,\Omega at 40∘C40^\circ\text{C}. What is the temperature coefficient of resistance of the material in K−1\text{K}^{-1}?

Answer: 0.004 K^-1

Answer

The temperature coefficient of resistance of the material is 0.004 K−10.004\,\text{K}^{-1} (or 4.0×10−3 K−14.0 \times 10^{-3}\,\text{K}^{-1}).
The temperature coefficient of resistance is calculated using α=RT−R0R0ΔT\alpha = \frac{R_T - R_0}{R_0 \Delta T}. Substituting R0=5.0 ΩR_0 = 5.0\,\Omega, RT=5.8 ΩR_T = 5.8\,\Omega, and ΔT=40 K\Delta T = 40\,\text{K} yields α=0.8200=0.004 K−1\alpha = \frac{0.8}{200} = 0.004\,\text{K}^{-1}.

Step-by-Step Solution

1
Identify the relationship between resistance and temperature.
RT=R0(1+αΔT)R_T = R_0(1 + \alpha \Delta T), with R0=5.0 ΩR_0 = 5.0\,\Omega, RT=5.8 ΩR_T = 5.8\,\Omega, and ΔT=40∘C\Delta T = 40^\circ\text{C}.
The resistance of metallic conductors varies linearly with temperature for moderate temperature changes.
2
Rearrange the expression to isolate the temperature coefficient α\alpha.
\alpha = \frac{R_T - R_0}{R_0 \Delta T}
Isolating α\alpha allows direct computation from the given resistance values and temperature interval.
3
Calculate the numerical value of α\alpha.
\alpha = \frac{5.8 - 5.0}{5.0 \times 40} = \frac{0.8}{200} = 0.004\,\text{K}^{-1}
Dividing the change in resistance by the product of initial resistance and temperature change gives the fractional resistance change per unit temperature change.

Key Concept

Temperature dependence of electrical resistance and temperature coefficient of resistance.
Rate this question

Topics

Electric Current and Resistance

More questions from this topic