Choose the correct relation, when the temperature of an...

Choose the correct relation, when the temperature of an isolated black body falls from $T_{1}$ to $T_{2}$ in time ' $t$ ', and assume ' $c^{\prime}$ to be a constant : (a) $t-c\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)$ (b) $t=c\left(\frac{1}{T_{2}^{2}}-\frac{1}{T_{1}^{2}}\right)$ (c) $t=c\left(\frac{1}{T_{2}^{3}} \ldots \frac{1}{T_{1}{ }^{3}}\right)$ (d) $t=c\left(\frac{1}{T_{2}{ }^{4}}-\frac{1}{T_{1}{ }^{4}}\right)$

Choose the correct relation, when the temperature of an isolated black body falls from $T_{1}$ to $T_{2}$ in time ' $t$ ', and assume ' $c^{\prime}$ to be a constant :
(a) $t-c\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)$
(b) $t=c\left(\frac{1}{T_{2}^{2}}-\frac{1}{T_{1}^{2}}\right)$
(c) $t=c\left(\frac{1}{T_{2}^{3}} \ldots \frac{1}{T_{1}{ }^{3}}\right)$
(d) $t=c\left(\frac{1}{T_{2}{ }^{4}}-\frac{1}{T_{1}{ }^{4}}\right)$

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Key Concepts

Thermal Cooling Dynamics in Isolated Systems

In isolated systems, the cooling process is governed by energy conservation where the loss of internal energy is balanced by the energy radiated away. This involves setting up and integrating a differential equation based on the radiated power from the system, which often leads to relations that connect temperature changes over time with a power law relating to temperature, as seen in the outcomes from the Stefan-Boltzmann law.

Blackbody Radiation

This concept refers to the idealized physical body that absorbs all incident electromagnetic radiation regardless of frequency or angle of incidence and re-emits energy as a blackbody spectrum that depends only on the body's temperature. It is a fundamental basis for understanding radiative properties in thermal physics.

Stefan-Boltzmann Law

This law explains that the total energy radiated per unit surface area of a black body in unit time is directly proportional to the fourth power of the body's absolute temperature. It is crucial for deriving cooling relations in radiating systems and establishes the strong temperature dependence of radiative energy loss.

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