Exams › JEE Advanced › Physics

A metal rod is heated by a wire that delivers constant power $P$. The rod is placed inside an insulated container, and its temperature $T$ varies with time $t$ as $T(t)=T_0(1+\beta t^4)$, where $\beta$ is a constant with suitable dimensions and $T_0$ is a constant representing temperature. What is the heat capacity of the metal rod?

1. $\dfrac{4P\big(T(t)-T_0\big)^3}{\beta T_0^4}$
2. $\dfrac{4P\big(T(t)-T_0\big)}{\beta T_0^4}$
3. $\dfrac{4P\big(T(t)-T_0\big)^4}{\beta T_0^3}$
4. $\dfrac{4P\big(T(t)-T_0\big)^2}{\beta T_0^3}$

Correct answer: $\dfrac{4P\big(T(t)-T_0\big)^2}{\beta T_0^3}$

Solution

For an insulated rod receiving constant power, the rate of heat gain is $P=C\,dT/dt$, where $C$ is the heat capacity. Differentiating the given temperature law and eliminating $t$ in favor of $T-T_0$ gives the required dependence. The resulting expression matches option D.

Related JEE Advanced Physics questions

• The relationship between the heat supplied $Q$ to a thermally isolated solid and its absolute temperature $\theta$ is given by $Q=a\theta^2+b\theta^3$, where $a$ and $b$ are constants. What is the expression for the heat capacity of the solid?
• Match the temperatures of a blackbody listed in Group I to the corresponding statement in Group II, and select the correct answer. Given: Wien’s constant = \(2.9\times10^{-3}\,\text{m·K}\) and \(hc/e = 1.24\times10^{-6}\,\text{V·m}\) Group I: (P) 2000 K (Q) 3000 K (R) 5000 K (S) 10000 K Group II: (1) The peak wavelength of emitted radiation can cause photoelectron ejection from a metal with a work function of 4 eV. (2) The peak wavelength of emitted radiation falls within the visible spectrum. (3) The peak wavelength of emitted radiation produces the broadest central diffraction maximum in a single-slit setup. (4) The energy radiated per unit area is one-sixteenth of that emitted by a blackbody at 6000 K. (5) The peak wavelength of emitted radiation is suitable for imaging human bones.
• A composite wall is built from two layers A and B of equal thickness but made of different materials. The thermal conductivity of layer A is three times that of layer B. In steady-state conditions, the total temperature difference across the entire wall is $36^\circ\text{C}$. What is the temperature difference across layer A alone?
• A steel rail track of length 1 km was laid at an ambient temperature of 20°C with no gaps for thermal expansion. When the temperature rose to 25°C, the track buckled and formed an isosceles triangle shape. Given the coefficient of linear expansion of steel is \(14\times10^{-6}\,\text{K}^{-1}\), find the height of the buckle in metres (to the nearest integer).
• Two rods are connected end to end. Rod 1 has length $l$ and thermal conductivity $2K$. Rod 2 has length $2l$ and thermal conductivity $K$. Both rods have the same cross-sectional area. What is the effective thermal conductivity of the combination?
• A steel container of water equivalent 10 g holds 20 g of ice at \(-30^\circ\text{C}\). Then 30 g of water at \(80^\circ\text{C}\) is poured into the container. Find the final equilibrium temperature. (Given: \(S_{\text{ice}}=0.5\ \text{cal g}^{-1}\,{}^\circ\text{C}^{-1}\), \(S_{\text{water}}=1\ \text{cal g}^{-1}\,{}^\circ\text{C}^{-1}\), \(L_{\text{ice}}=80\ \text{cal g}^{-1}\))

⚔️ Practice JEE Advanced Physics free + battle 1v1 →