More practice questions

4. Q4. At fixed emitting area and emissivity, the absolute temperature of a radiating body is doubled. By what factor does the power it radiates change? And if instead its temperature goes from 27 °C to 327 °C?

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   H ∝ T^⁴, so doubling T multiplies the power by 2^⁴ = 16. From 27 °C (300 K) to 327 °C (600 K) the absolute temperature also doubles, so the factor is again 16, not (327/27)^⁴.

5. Q5. A star’s spectrum peaks at 290 nm. Estimate its surface temperature, with b = 2.9 × 10^⁻³ m K.

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   T = b/λ_m = 2.9 × 10^⁻³/(290 × 10^⁻⁹), about 10 000 K. It looks bluish white, being hotter than the Sun.

6. Q6. A body of area 1.0 m^² and emissivity 0.5 is at 400 K in surroundings at 300 K. Find its net rate of heat loss by radiation.

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   H = eσA(T^⁴ − T_s⁴) = 0.5 × 5.67 × 10^⁻⁸ × 1.0 × (400^⁴ − 300^⁴), about 496 W.

7. Q7. Two blackbody spheres have radii r and 2r and temperatures 4000 K and 2000 K. Compare the powers they radiate.

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   H = AσT^⁴ with A = 4πr^², so H ∝ r^²T^⁴. The ratio is (r^² × 4000^⁴)/((2r)^² × 2000^⁴) = 16/4 = 4: the smaller, hotter sphere radiates 4 times as much.