A sphere, a cube and a thin circular plate, all of same material and same mass are initially heated to same high temperature.

The correct option is: Plate will cool fastest and sphere the slowest.

Step-by-Step Explanation:

According to Stefan-Boltzmann's law of radiation, the rate at which an object loses heat energy by radiation is given by:
$\frac{dQ}{dt}=e\sigma A(T^4-T_0^4)$
where:
• $e$ is the emissivity of the material,
• $\sigma$ is Stefan's constant,
• $A$ is the surface area of the body,
• $T$ is the temperature of the body, and
• $T{0}_{}$ is the temperature of the surroundings.

We also know that the heat lost by a body is related to its temperature fall by:
$dQ=mcdT$
where $m$ is the mass of the body and $c$ is its specific heat capacity.

Substituting this into the rate of heat loss equation, we get:
$mc\frac{dT}{dt}=e\sigma A(T^4-T_0^4)$
Solving for the rate of cooling $\frac{dT}{dt}$:
$\frac{dT}{dt}=\frac{e\sigma A(T^4-T_0^4)}{mc}$

Since the sphere, the cube, and the thin circular plate are all made of the same material and have the same mass $m$:
• Their emissivity $e$ and specific heat capacity $c$ are identical.
• Their initial high temperature $T$ and surrounding temperature $T{0}_{}$ are the same.
• Consequently, the rate of cooling depends only on their surface area $A$:
$\frac{dT}{dt}\propto A$

Since they all have the same material and the same mass, they must also have the same volume $V$ ($V=\frac{m}{\rho }$). For a given volume:
• A sphere has the minimum surface area of any geometric shape. Thus, $A_{\text{sphere}}$ is the smallest.
• A thin circular plate has the maximum surface area because it has two large flat circular faces and very small thickness. Thus, $A_{\text{plate}}$ is the largest.
• A cube has a surface area that lies between that of the sphere and the thin circular plate.

Since the rate of cooling is directly proportional to the surface area:
• The thin circular plate will cool at the fastest rate because of its large surface area.
• The sphere will cool at the slowest rate because it minimizes heat loss with the smallest surface area.