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A cylindrical conductor of resistance R is connected to an ideal battery of emf V. Initially at room temperature T0, the conductor loses heat to its surroundings at a rate proportional to the excess temperature (T - T0) with proportionality constant k. Mass and specific heat capacity are m and s respectively. Assuming resistance does not change with temperature, what is the temperature T of the conductor as a function of time t after connection?

1. T = T0 + (V² / (k*R)) * (1 - e^(-k*t / (m*s)))
2. T = T0 + (V² / (2*k*R)) * (1 - e^(-k*t / (m*s)))
3. After a long time, the temperature is T = T0 + V² / (k*R)
4. After a long time, the temperature is T = T0 + V² / (2*k*R)

Correct answer: T = T0 + (V² / (k*R)) * (1 - e^(-k*t / (m*s)))

Solution

The ODE gives theta(t) = (V²/(kR))*(1 - e^(-kt/(ms))), so T = T0 + (V²/(kR))*(1 - e^(-kt/(ms))). As t -> inf, T -> T0 + V²/(kR). Options A and C are correct.

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