A small spherical ball A of surface area 20 cm² is held at the centre of a hollow spherical shell B whose inner surface area is 80 cm². The surface of A and the inner surface of B behave as black bodies, the space between them is evacuated, and the shell B has very high thermal conductivit

Question

A small spherical ball A of surface area 20 cm² is held at the centre of a hollow spherical shell B whose inner surface area is 80 cm². The surface of A and the inner surface of B behave as black bodies, the space between them is evacuated, and the shell B has very high thermal conductivity. The heat capacities are 42 J/degC for A and 82 J/degC for B. At the instant when A is at 100 degC and B is at 20 degC, what is the approximate rate of change of temperature of ball A? (Take Stefan constant sigma = 5.67*10⁻⁸ W m⁻² K⁻⁴.)

Accepted Answer

-0.03 degC/s. Heat leaves A by radiation and is absorbed by B across the evacuated gap, so the only mode of transfer is radiation. Because both surfaces are black, the net radiative power emitted by A is P = sigma * A_A * (T_A⁴ - T_B⁴), where A_A is A's surface area. The temperature of A falls at rate -P divided by A's heat capacity. Plugging in A_A = 20*10⁻⁴ m², T_A = 373 K, T_B = 293 K gives P ~ 1.36 W, so dT_A/dt = -1.36/42 ~ -0.03 degC/s.

Solution

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