The given problem deals with finding the ratio of the amount of radiation radiated by a solid cylinder and a cube. Both the shapes have the same material, volume, and are heated up to the same temperature. After that, they are allowed to cool under similar conditions.




The amount of radiation radiated by a body depends on its surface area. Therefore, the first step is to calculate the surface area of the solid cylinder and the cube.

The surface area of a solid cylinder of length L and radius r is given by:
2πrL + 2πr^2

The surface area of a cube of edge length a is given by:
6a^2


The problem states that both the shapes are heated up to the same temperature. Therefore, we can assume that the temperature of both shapes is T.


The amount of radiation radiated by a body is given by the Stefan-Boltzmann law, which states that the amount of radiation radiated is proportional to the fourth power of the temperature and the surface area.

Therefore, the amount of radiation radiated by the solid cylinder is given by:
E1 = σT^4(2πrL + 2πr^2)

The amount of radiation radiated by the cube is given by:
E2 = σT^4(6a^2)


To find the ratio of the amount of radiation radiated by the solid cylinder and the cube, we divide E1 by E2.

(E1/E2) = [(2πrL + 2πr^2)/(6a^2)]


We can simplify the ratio further by using the fact that the volume of the cylinder and the cube is the same. Therefore, we have:

πr^2L = a^3

Substituting this in the ratio, we get:

(E1/E2) = [(2πrL + 2πr^2)/(6a^2)] = [(2πrL + 2πr^2)/(6π(a^2)(a/π))] = [(r/L) + (r^2/a^2)]/(a/π)


Thus, the ratio of the amount of radiation radiated by the solid cylinder and the cube is given by:
(E1/E2) = [(r/L) + (r^2/a^2)]/(a/π)


The ratio of the amount of radiation radiated by a solid cylinder and a cube depends on the ratio of the radius to the length of the cylinder and the ratio of the radius to the edge length of the cube. If both shapes have the same volume and are heated up to the same temperature, then the ratio of the amount of radiation radiated can be calculated using the