A metal wire has a resistance of 8.00Ω at a temperature of 20C. If the same wire has a resistance of 8.40Ω at 70C, what is the resistance of this wire in ohms when its temperature is −10C?
R₀=8 Ω, T₀=20℃,
R₁=8.4 Ω, T₁=70℃,
T₂ = - 10℃
R₂=?
R=R₀[1-αΔT]
R₁=R₀[1-αΔT₁]
8.4=8[1+(70-20)α]
α={8.4/8 – 1)/50 = 10⁻³ K⁻¹
R₂=R₀[1-αΔT₂] =
=8[1+ 10⁻³•(-10-20)] =7.76 Ω
To solve this question, we need to use the temperature coefficient of resistance. The resistance of a wire changes with temperature, and this change can be described by the temperature coefficient. The temperature coefficient of resistance is usually denoted by the symbol α.
Given:
Resistance at 20°C (R₁) = 8.00 Ω
Resistance at 70°C (R₂) = 8.40 Ω
Temperature at which resistance is required (T₃) = -10°C
Step 1: Calculate the temperature change
ΔT = T₂ - T₁
ΔT = 70°C - 20°C
ΔT = 50°C
Step 2: Calculate the temperature coefficient of resistance
Using the formula:
α = (R₂ - R₁) / (R₁ * ΔT)
α = (8.40 Ω - 8.00 Ω) / (8.00 Ω * 50°C)
α = 0.40 Ω / (8.00 Ω * 50°C)
α = 0.40 Ω / 400 Ω°C
α = 0.001 Ω/°C
Step 3: Calculate the resistance at the desired temperature (T₃)
Using the formula:
R₃ = R₁ * (1 + α * ΔT)
R₃ = 8.00 Ω * (1 + 0.001 Ω/°C * (-10°C))
R₃ = 8.00 Ω * (1 - 0.01)
R₃ = 8.00 Ω * 0.99
R₃ ≈ 7.92 Ω
Therefore, the resistance of the wire at a temperature of -10°C is approximately 7.92 Ω.