The Rayleigh-Jeans distribution function is I(2,T)= 2πckgT 24 where kB is the Boltzmann constant. (i) Sketch the two distribution functions to show the approximation to Planck's formula for long wavelength. (ii) Explain why it is expected that the classical result is much better at long wavelength?
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- In a supercollider at CERN, protons are accelerated to velocities of 0.25c. What are their wavelengths at this speed? What are their kinetic energies? If a beam of protons were to gain its kinetic energy in only one pass through a potential difference, how high would this potential difference have to be? (Rest mass energy of a proton is E0=938 MeV).1. (Kittel 6.4) Energy of gas of extreme relativistic particles. Extreme relativistic particles have momenta p such that pc >> Mc², where M is the rest mass of the particle. The de Broglie relation λ = h/p for the quantum wavelength continues to apply. Show that the mean energy per particle of an extreme relativistic ideal gas is 3 37 if & pc in contrast to for the nonrelativistic problem. (An interesting variety 2 of relativistic problems are discussed by E. Fermi in Notes on Thermodynamics and Statistics, University of Chicago Press, 1966, paperback.)A) What is the approximate wavelength emitted from helium represented by the bright yellow emission line below? What is it's frequency in HZ and energy in eV? (1 eV= 1.6 x 10-19 joules). B) If the excited helium electron that emits a yellow photon in this line starts with a potential energy of 8 eV, what is the potential energy of the electron afterwards? Assume that the emission of a yellow photon is allowed by the laws of quantum mechanics. Also don't worry about the other electron.
- Imagine an alternate universe where the value of the Planck constant is 6.62607x10−4J·s. In that universe, which of the following objects would require quantum mechanics to describe, that is, would show both particle and wave properties? Which objects would act like everyday objects, and be adequately described by classical mechanics? A grain of sand with a mass of 135 mg, 515. µm wide, moving at 4.00 mm/s. An airplane with a mass of 1.75 x 104 kg, 15.0 m long, moving at 2300. km/h. An atom with a mass of 1.0 x 10-27 kg, 137. pm wide, moving at 394. m/s. A ball with a mass of 215. g, 4.1 cm wide, moving at 35.0 m/s.1.the electrons which are initially at rest are accelerated through a potential difference of 125V.calculate the de Broglie wavelength. 2.two long parallel wires carrying currents i1 and i2 in opposite directions. what are the magnitude and direction of the net magnetic field at point P ? ASSUME THE FOLLOWING VALUES i1=15A,i2=32A, and d=5.3cm.answer only part d) 2. a) Show that AE/E, the fractional change in energy of a photon in a Compton effect, is equal to (hv' /moc²)(1– cos0). b) Plot AE/E against 0 and interpret the curve physically. c) What is the fractional increase in wavelength that leads to a 75% loss of photon energy in a Compton collision? d) Through what angle must a 0.2 MeV photon be scattered by a free electron in order for it to lose 10% of its energy?
- 1. Compare the λmax (ex) with λmax (em). If there is a ΔE between the ground state and the excited state, why is the excitation λmax different than the fluorescence emission λmax. What is this displacement called? 2. Explain why it is more convenient to quantify by selecting the maximum absorption / emission wavelengths?3. For Blasius solution of laminar boundary layer, prove for the momentum thickness, U "V 2vx f'(1 – f') dn = f"(0) = 0.4696 0.664 V Re. |3. Dimensional analysis can provide insight into Stefan-Boltzmann's law for the radiation from a black body. According to this law the intensity of radiation, in units of J s-' m-², from a body at temperature Tis 1 = GT*, where e is Stefan-Boltzmann's constant. Because black-body radiation can be considered to be a gas of photons, i.e. quantum particles which move with velocity e with typical energies of the order of kT, the intensity I is a function of h, c and kT. Use dimensional analysis to confirm that Iis proportional to 7 and find the dependence of a on h and c.
- .5. Imagine a universe in which you have a de Broglie wavelength of 1 meter. In your everyday life, you would then be susceptible to experience considerable wave-like phenomena, such as diffraction and interference. Estimate the speed you would need to have in order to acquire this wavelength. Also, compute how long it would take you to move through 0.8 meter (i.e., about one step) with this speed.1. We see light from a distant galaxy, galaxy “A”, with a redshift of z = 0.017. What wavelengths would we observe from the Hydrogen atom transitions from n = 5 to n = 2, from n = 4 to n = 2, and from n = 3 to n = 2? 2. Building on the previous problem, imagine there is a galaxy “B” that is in the same direction as galaxy "A", but "B" is at a redshift of z = 0.034 as seen from our Galaxy. What if the astronomers in galaxy “B” were observing galaxy “A”.... What wavelengths would they observe for these same Hydrogen transitions in galaxy “A”? (Hint: You should not have to do any calculations to answer this question, but please explain your answer carefully.)Answer= 11. Calculate the uncertainty in the speed of a ball of mass 500 g that is known to be within 1.0 um of a certain point on a bat. 12. The Planck distribution gives the energy in the wavelength range da at the wavelength A. Calculate the energy density in the range fro 650 nm to 655 nm inside a cavity of volume 100cm³" when the temperatura in 25 C