A thermally insulated piece of metal is heated under...

A thermally insulated piece of metal is heated under atmosphere by an electric current so that it receives electric energy at a constant power $P$. This leads to an increase of the absolute temperature $T$ of the metal with time $t$ as follows $T=a t^{1 / 4}$ Then the heat capacity $C_{p}$ is : (a) $\frac{4 P T^{5}}{a^{4}}$ (c) $4 P T^{2}$ (b) $\frac{4 P T^{\overline{4}}}{a^{3}}$ (d) none of these

A thermally insulated piece of metal is heated under atmosphere by an electric current so that it receives electric energy at a constant power $P$. This leads to an increase of the absolute temperature $T$ of the metal with time $t$ as follows $T=a t^{1 / 4}$
Then the heat capacity $C_{p}$ is :
(a) $\frac{4 P T^{5}}{a^{4}}$
(c) $4 P T^{2}$
(b) $\frac{4 P T^{\overline{4}}}{a^{3}}$
(d) none of these

Key Concepts

Relating Energy, Temperature Change, and Time

With the expression T = a·t^(1/4) and the constant power input, one can derive the expression for the heat capacity by first differentiating the temperature function with respect to time, then rearranging the equation P = C_p · (dT/dt) to solve for C_p. This process connects the concepts of energy input, time evolution of the temperature, and the material's heat capacity.

Using the Chain Rule for Differentiation

When temperature is given as a function of time by a power law (such as T = a·t^(1/4)), the chain rule is employed to differentiate this function with respect to time. This step is essential for finding the rate of temperature change dT/dt, which directly relates to the heat capacity through the equation P = C_p · (dT/dt).

Heat Capacity

Heat capacity is a measure of how much energy is required to raise the temperature of a substance by a given amount. In thermodynamics, it is defined as C_p = dQ/dT, where dQ is the amount of heat added and dT is the resulting temperature change. This concept allows us to link the energy input to the temperature response of the material.

Constant Power Heating

When a system is heated at constant power, a fixed amount of energy is delivered per unit time. This means the rate of heat input is constant (P = dQ/dt), and understanding this relationship is key to relating the energy supplied over time to the temperature change of the system.

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