University Physics with Modern Physics (14th Edition)

Published by Pearson
ISBN 10: 0321973615
ISBN 13: 978-0-32197-361-0

Chapter 39 - Particles Behaving as Waves - Problems - Exercises - Page 1315: 39.41


a. 1.7 T. b. 0.58.

Work Step by Step

The stars behave like blackbodies and obey the Stefan-Boltzmann law, and Wien’s displacement law. The Stefan-Boltzmann law says that the intensity of the radiation is proportional to the fourth power of the absolute temperature, so the total radiated power is $P=\sigma A T^4$. a. The hot and cool stars each radiate the same total power, so the Stefan-Boltzmann law gives this: $$\sigma A_c T_c^4=\sigma A_h T_h^4$$ The cooler star has 3 times the radius of the hotter star. $$\sigma 4 \pi(3R_h)^2 T_c^4=\sigma 4 \pi (R_h)^2 T_c^4$$ $$9 T_c^4=T_h^4$$ $$T_h=1.7 T_c$$ We rounded $\sqrt{3}$ to 2 significant figures. b. Using Wien’s law, take the ratio of the peak wavelengths. $$\frac{\lambda_{m,hot}}{\lambda_{m,cool}}=\frac{T_c}{T_h}=\frac{1}{\sqrt{3}}=0.58$$
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