A solid cylinder of length L and radius r is heat upto same temperature as that of a cube of edge length a. If both have same material, volume and allowed to cool under similar conditions, then ratio of amount of radiations ratiated will be ( Neglected ratiated emitted from flat surface of the cylinder) (1) a/3r (2) 2a/rL (3) a^2/rL (4) πa/2rL?

Question

Answer

Answer:

Introduction:
In this question, we have to find the ratio of the amount of radiation radiated by a solid cylinder and a cube of the same material, volume, and under the same conditions. The cylinder is heated up to the same temperature as that of the cube.

Formula:
The radiation emitted by a body is given by the Stefan-Boltzmann law, which states that the radiation emitted per unit time per unit area is proportional to the fourth power of the absolute temperature of the body.

The formula for the radiation emitted by a body is given as:
E = σAT^4
where E is the energy radiated per unit time per unit area, σ is the Stefan-Boltzmann constant, A is the surface area of the body, and T is the absolute temperature of the body.

Solution:

Step 1: Calculation of Surface Area of Cylinder
The surface area of the cylinder is given by:
A = 2πrh + 2πr^2
where h is the height of the cylinder and r is the radius of the cylinder.

As the cylinder is solid, its height is equal to its length.
Therefore, h = L.

So, A = 2πrL + 2πr^2

Step 2: Calculation of Surface Area of Cube
The surface area of the cube is given by:
A = 6a^2
where a is the edge length of the cube.

Step 3: Calculation of Radiation Emitted by Cylinder
The energy radiated per unit time per unit area by the cylinder is given by:
E = σAT^4

As the cylinder and the cube are heated up to the same temperature, their absolute temperatures are the same.
Therefore, we can take T^4 as a common factor.

So, E = σT^4(2πrL + 2πr^2)

Step 4: Calculation of Radiation Emitted by Cube
The energy radiated per unit time per unit area by the cube is given by:
E = σAT^4

As the cylinder and the cube are heated up to the same temperature, their absolute temperatures are the same.
Therefore, we can take T^4 as a common factor.

So, E = σT^4(6a^2)

Step 5: Calculation of Ratio of Radiation Emitted
The ratio of the amount of radiation radiated by the cylinder to the amount of radiation radiated by the cube is given by:
E(cylinder)/E(cube) = [σT^4(2πrL + 2πr^2)] / [σT^4(6a^2)]

On simplifying, we get:
E(cylinder)/E(cube) = (a^2)/(3rL)

Therefore, the correct option is (1) a/3r.

Conclusion:
Hence, we can conclude that the ratio of the amount of radiation radiated by the cylinder to the amount of radiation radiated by the cube is (1) a/3r.

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