Physics 103 Recitation, November 6, 2007 Planck constant: h = 6.64 × 10−34 J s = 4.14 × 10−15 eV s m Speed of light: c = 3.0 × 108 s Planck constant times the speed of light: hc = 1240 eV nm −13.6 eV Energy levels of hydrogen atom: E = n n 2 Electron volt: 1 eV = 1.602 × 10−19 J Photon energy: E = hf c hc 1240 eV nm Photon wavelength: = = = f E E
Spectrum of visible light:
Some problems to work....
1. The figure below shows the energy level diagram of a recently discovered element, Brynmawrium. If a sample of Brynmawrium gas is excited (e.g., a high voltage is put across it), at what wavelengths will it emit visible light?
E4=−4eV
E3=−13eV
E2=−15eV E1=−16eV
2. What is the longest wavelength of light that Brynmawrium can absorb?
More on the other side.... 3. The energy level diagram for the Rydberg model of hydrogen atoms, powerful though it is, is an oversimplification. One thing it neglects each energy level consists of several di erent states. There are the s, p, d, f, etc., orbitals that you may remember from chemistry. There are restrictions on how exactly an electron can jump from one type of orbital to another. Another thing our simple hydrogen model neglects is hyperfine structure, which results from a very small di erence in the energy level of the atom depending on whether the “spin” of the electron and nucleus are aligned. “Spin” is a quantum property of particles which is vaguely related to rotation. It is a vector quantity, so it “points” in some direction. If the spins of the electron and the proton are aligned (pointing in the same direction), the atom has slightly more energy than if they are anti-aligned. (They are either aligned or anti-aligned; no other options are possible. Isn’t quantum physics bizarre?) An atom going from the aligned state to the anti-aligned state emits light with wavelength 21.106 cm.
(a) What is the di erence in hyperfine energy levels, in eV? (b) If you compare two atoms dropping from the n = 2 state to the n = 1 state, and one ends up in each of the hyperfine levels of the n = 1 state, what is the di erence in the wavelengths of the two photons emitted?
4. The intensity of sunlight hitting the ground is around 1400 W/m2 when the Sun is directly overhead. Estimate (within a factor of a few) the number of photons hitting a 1 m2 patch of ground in 1 second.
5. This is a variation on a problem we did last week. Imagine two radio transmitters which emit exactly the same signal at the same time. They are separated on an east-west line by some distance d. They emit waves of wavelength .
(a) In what directions is the signal strongest (constructive interference)? Give your answer as a formula for an angle, , relative to due east. (b) What are the numerical values of for = 280 m and d = 700 m? (c) What would happen if there weren’t just two antennas, but a long series of antennas spaced at intervals of d?