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P316 Fall 2006 1. From Planck’s Law Homework #1 3 k BT ( , T ) 2 3 e 1 c derive Wien’s Law for the wavelength 1 , max at which the intensity of the blackbody spectrum is maximum, i.e. Hint: e x 1 / 1 x / 3 max T Const. has a single non-zero solution. 2. Derive Planck’s law in terms of wavelength instead of angular frequency . 3. Show that the total electromagnetic energy density in a cavity with walls at constant temperature T (an oven) is proportional to T4 x3 4 Hint: e 0 4. 5. 6. 7. x dx 15 1 and find the proportionality constant (it is called the Stefan-Boltzmann Constant). When light with =450 nm shines on Potassium, photoelectrons with stopping potential of 0.52V are emitted. If the wavelength of incident light is changed to =300 nm, the stopping potential becomes 1.9V. Find the work function of potassium and compute the value of Planck’s constant. If the angular momentum of the Earth’s orbit around the Sun were quantized what would the Earth’s quantum number be? How much energy would be released in a transition to the next lower quantum number? Would this amount of energy be detectable? (Earth orbit is 1.5x1011 m radius). Suppose the nucleus has charge Z(+e) and a single electron orbits it as described by Bohr. Write down equations for the radii of orbits of each quantum state n, the energies of each state, and the general equation for the wavelengths of emitted light. Light from Sun arrives at Earth at average rate 1.5 kWatt/m2. (a) Assume the light is monochromatic at frequency3x1015 Hz. How many photons per second hit the earth? What is the power output of the sun and the number of photons per second it emits? (b) If we model a spaceship as an aluminum sphere of radius 10m (Al has work function 4 Volts), what electric potential will be developed by it due to the photoelectric effect? How much charge is on the spaceship?