The space between two hollow concentric spheres of radii 0.1 m and 0.2 m is under vacuum.
Exchange of radiation (uniform in all directions) occurs only between the outer surface (S₁) of the smaller sphere and the inner surface (S₂) of the larger sphere. The fraction (rounded off to the second decimal place) of the radiation energy leaving S_{2 .}Whcih reaches S₁is ____________.
Correct answer is '0.125'. Can you explain this answer?

Question

The space between two hollow concentric spheres of radii 0.1 m and 0.2 m is under vacuum.
Exchange of radiation (uniform in all directions) occurs only between the outer surface (S₁) of the smaller sphere and the inner surface (S₂) of the larger sphere. The fraction (rounded off to the second decimal place) of the radiation energy leaving S_{2 .}Whcih reaches S₁is ____________.

Correct answer is '0.125'. Can you explain this answer?

Answer

Explanation:

The radiation exchange between two concentric spheres is given by the formula:

q = A1 σ (T1^4 - T2^4)

where q is the rate of radiation exchange, A1 is the surface area of the inner sphere, σ is the Stefan-Boltzmann constant, T1 is the temperature of the inner sphere, and T2 is the temperature of the outer sphere.

Since the space between the spheres is under vacuum, there is no convection or conduction of heat, and the temperatures of the spheres will be uniform.

Let T be the common temperature of the spheres.

The energy radiated per unit time per unit area from S1 is given by:

q1 = A1 σ T^4

The energy absorbed per unit time per unit area by S2 is given by:

q2 = A2 σ T^4

where A2 is the surface area of the outer sphere.

The fraction of energy leaving S2 that reaches S1 is given by:

q1/q2 = A1/A2 = (0.1/0.2)^2 = 0.25

Rounding off to two decimal places, we get the fraction as 0.25, which is equivalent to 0.125 in percentage.

Therefore, the fraction of the radiation energy leaving S2 that reaches S1 is 0.125.

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