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Question #42149

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The Question

A heating coil of resistance $R$ and inductance $L$ is immersed in a calorimeter of heat capacity $C$ . At $t=0$ , the coil is connected to an ideal DC voltage source $V_0$ . The calorimeter loses heat to the surroundings (at constant temperature $T_0$ ) at a rate $k(T - T_0)$ , where $T$ is the instantaneous temperature of the calorimeter and $k$ is a positive constant. Assume the electrical time constant of the coil is exactly equal to the thermal time constant of the calorimeter, i.e., $\frac{L}{R} = \frac{C}{k} = \tau$ . If the initial temperature of the calorimeter is $T_0$ , which of the following expressions correctly gives the temperature difference $\Delta T(t) = T(t) - T_0$ at any time $t > 0$ ?

Options

A.
$\frac{V_0^2}{k R} \left[ 1 - \frac{2t}{\tau} e^{-t/\tau} - e^{-2t/\tau} \right]$
B.
$\frac{V_0^2}{k R} \left[ 1 - e^{-t/\tau} \right]$
C.
$\frac{V_0^2}{k R} \left[ 1 - \left(1 + \frac{t}{\tau}\right) e^{-t/\tau} \right]$
D.
$\frac{V_0^2}{k R} \left[ 1 - \frac{t}{\tau} e^{-t/\tau} - e^{-2t/\tau} \right]$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Calorimetry with Heat Loss|Transient Current in RL Circuit|Work, Energy, and Power

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of transient electrical circuits and non-steady-state calorimetry.

The lethal trap lies in the non-linear coupling of the transient current to the thermal power ( $P \propto I^2$ ).

Students often incorrectly assume the power grows linearly with the current transient, leading to $P \propto (1 - e^{-t/\tau})$ (Option C), or they completely ignore the transient phase and assume steady-state power $V_0^2/R$ from $t=0$ (Option B).

By equating the electrical and thermal time constants, the problem forces the resolution of a first-order non-homogeneous differential equation with a resonant-like forcing term, rigorously testing the ability to set up and integrate coupled physical rate equations without falling for algebraic shortcuts.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Thermal Properties of Matter - Class 11 Part 2
Concept	Calorimetry with Heat Loss
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free