Two spheres A and B have same radius but heat capacity of A...

Two spheres $A$ and $B$ have same radius but heat capacity of $A$ is greater than that of $B$, The surfaces of both are painted black, They are heated to same temperature and allowed to cool then initially (a) $A$ cools faster than $B$ (b) Both $A$ and $B$ cool at the same rate (c) Energy radiated in unit time from unit area is same for both the spheres (d) $B$ cools faster than $\Lambda$

Two spheres $A$ and $B$ have same radius but heat capacity of $A$ is greater than that of $B$, The surfaces of both are painted black, They are heated to same temperature and allowed to cool then initially
(a) $A$ cools faster than $B$
(b) Both $A$ and $B$ cool at the same rate
(c) Energy radiated in unit time from unit area is same for both the spheres
(d) $B$ cools faster than $\Lambda$

Key Concepts

Stefan-Boltzmann Law

The Stefan-Boltzmann law governs the power radiated from a black body in terms of its temperature, stating that the energy emitted per unit area is proportional to the fourth power of the absolute temperature. This law means that two objects with identical surface areas and temperatures will radiate the same amount of energy, independent of their internal properties such as heat capacity.

Black Body Radiation

Black body radiation refers to the electromagnetic radiation emitted by an idealized object that absorbs all incident radiation and re-emits energy perfectly. In the context of this problem, since both spheres are painted black (ensuring they behave closely to perfect black bodies), they will emit energy according to the Stefan-Boltzmann law, making the radiative power per unit area identical for both spheres when at the same temperature.

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to change the temperature of a substance by a given amount. Although the two spheres radiate energy at the same rate (per unit area), the sphere with a lower heat capacity will experience a larger temperature drop for the same amount of energy lost, causing it to cool faster than the sphere with a higher heat capacity.

Thermal Inertia

Thermal inertia, often associated with the concept of thermal mass, is the resistance of an object to a change in temperature when heat is added or removed. A higher heat capacity indicates higher thermal inertia, meaning the object will change temperature more slowly. In contrast, an object with lower thermal inertia (lower heat capacity) will show a quicker temperature response to energy loss.

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