The resistance of an electrical toaster has a temperature dependence given by R(T)=R₀[1+α(T-T₀)] in its range of operation. At T₀=300 K, R=100 Ω and at T=500 K, R=120 Ω. The toaster is connected to a voltage source at 200 V and its temperature is raised at a constant rate from 300 to 500 K in 30 s.…

Question

The resistance of an electrical toaster has a temperature dependence given by R(T)=R₀[1+α(T-T₀)] in its range of operation. At T₀=300 K, R=100 Ω and at T=500 K, R=120 Ω. The toaster is connected to a voltage source at 200 V and its temperature is raised at a constant rate from 300 to 500 K in 30 s. The total work done in raising the temperature is J. (Nearest integer)

Accepted Answer

10939. From the data: 120=100[1+α(200)]⇒α=10⁻³ K⁻¹. Constant heating rate: (dT)/(dt)=(200)/(30) K s⁻¹, so dt=(30)/(200)dT. W=∫(V²)/Rdt=(V²)/(R₀)·(30)/(200)∫₃₀₀⁵⁰⁰(dT)/(1+α(T-300)). W=400×0.15×1/α ln(1.2)=60000 ln 1.2 J. W≈10939 J. The book marks none of its options correct. They were 400 ln 5/6 J, 200 ln 2/3 J, 300 J and 400 ln (1.5)/(1.3) J; option (d) is only (V²)/(R₀)ln (1+500α)/(1+300α) (absolute T used in the bracket and the factor (Δ t)/(αΔ T)=150 s dropped), so this is set as a numerical question.

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