A coil of copper wire has a resistance of 40 W at 20°C and the temperature coefficient of the wire is 0.004 /°C at 20°C. Determine the temperature of the wire when the resistance is 46 W.
Question
A coil of copper wire has a resistance of 40 W at 20°C and the temperature coefficient of the wire is 0.004 /°C at 20°C. Determine the temperature of the wire when the resistance is 46 W.
Solution
To solve this problem, we can use the formula for resistance change with temperature, which is:
Rt = R0 * (1 + α * (T - T0))
where:
- Rt is the resistance at temperature T (46 Ω in this case),
- R0 is the resistance at reference temperature T0 (40 Ω at 20°C in this case),
- α is the temperature coefficient of resistance (0.004 /°C in this case).
We need to find T, so we rearrange the formula to solve for T:
T = ((Rt / R0) - 1) / α + T0
Substituting the given values:
T = ((46 Ω / 40 Ω) - 1) / 0.004 /°C + 20°C
T = (1.15 - 1) / 0.004 /°C + 20°C
T = 0.15 / 0.004 /°C + 20°C
T = 37.5°C + 20°C
T = 57.5°C
So, the temperature of the wire when the resistance is 46 Ω is 57.5°C.
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