Physics 220 Homework #1 Spring 2016 Due Wednesday, April 6, 2016 1. Using Excel or Mathematica or another plotting software, plot the thermal spectrum (the Rayleigh-Jeans law with Planck’s correction) for a temperature of T = 5900K . What feature(s) does your graph show? What happens to the maximum wavelength of emission from the blackbody as the temperature of the object increases? Is this as expected? What does the area under each curve represent? Now on the same plot, add just the Rayleigh-Jeans law (without Planck’s correction) for a temperature of T = 5900K . What do you notice? (For this question provide the fully commented code from the software package that you used along with the graph that you produce.) 2. We derived the Rayleigh-Jeans law with Planck’s correction in terms of the frequency of light. What does the Rayleigh-Jeans law with Planck’s correction look like in terms of the wavelength of light? 3. The rate at which energy from the Sun reaches a unit area on the Earth is called the solar constant ( S ). The solar constant is measured to be S = 1350 mW2 on the Earth at a distance rE→S = 1.5 × 1011 m . What is the energy output of the Sun? If the Sun has an average radius of RSun = 6.96 × 10 8 m , what is the approximate surface temperature of the Sun? 4. A light bulb emits 40W of energy every second at a wavelength of 6 × 10 −7 m . What is the total number of photons emitted every second and what is the energy of one of these photons? 5. Estimate the temperature of a “white” hot light bulb filament. If the light bulb has a radiated power of 100W , what is the surface area of the filament? How many photons are radiated per second? 6. A thermal light source has a temperature of T = 6000K and a total radiated power of dS 100W . What is the power per unit area per µ m ( ) at the peak of the thermal dλ spectrum? 7. Suppose that the Earth was formed as molten rock at some very high temperature. a. About how much time would it take the Earth to cool to say 300K ? The radius of the Earth is 6.4 × 10 6 m and the average kinetic energy of a particle in the Earth 3 is given by E = k BT . 2 b. Is this time consistent with what we know for the age of the earth ( ~ 4.5 × 10 9 yr )? If this time is not consistent with the age of the earth, what could explain the difference? c. Speaking of the Earth, there are some crazy things that go on. Suppose that, for some strange reason, a naked person at the South Pole gets out of a sauna (hot tub) with a body temperature of 311K . The person encounters the outside air, which is at a temperature very much below the freezing point of water. How much energy per second does the person lose by radiation cooling if their surface area is 1m 2 ? d. By approximately what factor would this person’s loss of energy to radiation cooling differ from the previous case if the person now were to step out of the sauna into a cool room with a temperature of 290K ?