Suspended inside a closed box maintained at 57^ C , a spherical ball with a surface area of 20 cm ^2 absorbs all incident radiation. Determine the net rate of heat flow to or from the ball at an instant when its temperature is 200^ C . Use a Stefan constant of 6.0 10⁻⁸ W m ⁻² K ⁻⁴ .

Step-by-step solution

The temperature of the ball is T = 200^ C = 473 K . The temperature of the surroundings is T₀ = 57^ C = 330 K . The surface area of the ball is A = 20 cm ^2 = 20 10⁻⁴ m ^2 . Since the ball absorbs all incident radiation, it behaves as a perfectly black body, giving an emissivity e = 1 . According to the Stefan-Boltzmann law, the net rate of heat flow is given by: P = e A (T^4 - T₀^4) Substituting the given values: P = (6.0 10⁻⁸) 1 (20 10⁻⁴) (473^4 - 330^4) P = 12 10⁻¹¹ (5.005 10¹⁰ - 1.186 10¹⁰) P = 12 10⁻¹¹ (3.819