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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?