Asphere is originally at a temperature of 500°c. the sphere is melted and recast, without loss of mass, into a cube with the same emissivity as the sphere. calculate the celsius temperature of the cube if it is to emit the same radiant powers the sphere

The value of the celsius temperature of the cube is 472.2°c.

Explanation:        

The expression for the power radiated is as follows;

Here, A is the area, is the stefan's constant, is the emissivity and T is the temperature.

It is given in the problem that A sphere is originally at a temperature of 500°c. The sphere is melted and recast, without loss of mass, into a cube with the same emissivity as the sphere.

Then the expression for the radiated power for the cube and the sphere can be expressed as;

Here, is the area of the sphere, is the area of the cube,  is the temperature of the sphere and  is the temperature of the cube.

The radiated powers and emissivity of the cube and the sphere are same.

The area of the sphere is .

Here, r is the radius of the sphere.

The area of the cube is .

Here, a is the edge of the cube.

Put and .

 ....(1)

The masses and the densities of the sphere and the cube are same. Then the volumes are also same.

Here, are the volumes of the sphere and the cube.

Put this value in the equation (1).

Put T_{1}=500°c.

Therefore, the value of the celsius temperature of the cube is 472.7°c.