PHAS0011 - Modern Physics, Astronomy and Cosmology 2021
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PHAS0011 - Modern Physics, Astronomy and Cosmology
1. (a) In the photoelectric effect the metal’s work function is 5.0 eV.
What is the frequency of the photon that has the minimal
energy needed to eject a photoelectron from this metal?
(b) Determine the wavelength of a photon necessary to have:
(i) The same kinetic energy E as a 60 g tennis ball travelling at 230 km hour-1 ;
(ii) The same momentum p as a 60 g tennis ball travelling at 230 km hour-1 ;
(iii) The same momentum p as a 1 mg feather travelling at 0.05 km hour-1 .
(c) Consider the version of Heisenberg’s Uncertainty Principle that relates energy and time.
(i) What are the minimum uncertainties in energy (in J and eV) of states that last for 1 ps and 2 s?
(ii) What are the minimum lifetimes of states that have uncertainties in energy 1 J and 1 eV?
(d) A particle with mass m in some potential U(x) has energy equals to zero and time-independent wave function
w(x) = Axe-x2 /L2 ,
where A and L are constants.
(i) By using the time-independent Schrdinger equation determine the potential energy U(x) of the particle. (ii) Determine the value of x at the minimum of U(x) and the value of U there.
(e) (i) A particle has a 1-dimensional wave function of the form
w(x) = C for a s x s 4a
and w(x) = 0 elsewhere.
Find that value of C that normalises w.
(ii) What are the dimensions of C?
(iii) What is the probability of finding the particle in the range
a s x s 3a?
(f) For the Hydrogen atom, the radial solutions of the
3-dimensional time-independent Schrdinger equation for the largest possible angular momentum for each value of the principal quantum number n are
w(r) = An rn-1 e-r/(na0 ) ,
where An and a0 are constants.
Calculate the more probable radii and show that they are
quantised.
2. (a) Describe briefly how the Doppler effect has been used to detect planets around distant stars.
A star of 1 Mo has a planet of Jupiter’s mass orbiting in a circular orbit of radius 0.30 A.U. and period of 60 days. If the system is viewed edge on to the planet’s orbit, what would be the total wavelength shift in the Balmer Hα absorption line in the star’s spectrum due to its orbital motion? What spectral
resolution would be required to detect this planet? (The rest
frame wavelength of Hα is 656.2801 nm).
(b) Describe two distinct observations of galaxies that have
indicated the presence of dark matter in the universe and, for
each case, discuss the physical basis of how the relative proportions of dark and visible matter can be estimated.
How have observations of the motions of stars in the Milky Way been used to demonstrate that this dark matter cannot lie in the Galactic disk?
(c) Describe briefly how, during the first three minutes of cosmic history, the nuclei of certain light elements were synthesised and why the primordial abundances of these elements provide independent evidence that dark matter cannot be baryonic material. Since helium is one of these elements and also synthesised in stars, how might astronomers determine
its primordial abundance?
(d) A star is in its helium burning phase on the horizontal branch where the helium is fused to produce carbon by the triple alpha process as shown below:
3 He4 → C12
(i) What is the mass deficit and the energy released in the
above reaction? (ii) The star has a luminosity of 100Lo and had an initial main sequence mass of 1.5Mo and 10% of that initial mass was converted to helium. For how long in years does the star
remain on the horizontal branch?
(e) Describe clearly the physical processes that are believed to
lead to the explosions of Type I and Type II supernovae? Which type might be used as an indicator of the current star formation rate in a galaxy and why? Which type would be a valuable distance indicator for cosmological purposes and
why?
(f) Gravitational lensing was predicted by Einstein and verified
by Eddington who measured the deflection of starlight close to the sun at the time of a total eclipse. Since eclipses are rare, Einstein first wrote to the Californian astronomer, George Ellery Hale, and inquired whether it was practical to measure the deflection of starlight using the planet Jupiter. Estimate the deflection angles expected for the Sun and
Jupiter and comment on the practicality of Einstein’s request.
(g) Remarkably, neither Einstein nor Eddington considered
gravitational lensing would be useful in astronomy beyond verifying General Relativity. Describe briefly three cases where the use of gravitational lensing has been important in shaping our view of the universe and the evolution of
galaxies.
3. (a) A small child arrives at a hospital following a serious accident during which it is suspected that a thin steel needle has become embedded somewhere inside the child’s foot, either within the heel bone or within the surrounding soft tissue. Briefly describe any advantages and disadvantages of using the following medical imaging techniques to confirm the presence of the needle and determine its location prior to surgical removal.
(i) diagnostic ultrasound
(ii) x-ray imaging
(iii) x-ray computed tomography
(iv) magnetic resonance imaging.
Figure 1
(b) As illustrated in Figure 1, a beam of x-rays passes through a
uniform thickness of soft tissue which contains a small region
of bone. The linear attenuation coefficients of soft tissue and
bone are 0.015 mm-1 and 0.065 mm-1 respectively. The attenuations of the beam measured at points A and B are 3.9
dB and 6.1 dB respectively. Use this information to calculate the following:
(i) the thickness of the bone d, (ii) the contrast produced by the bone, where contrast is
defined as equal to the difference between the intensities measured at A and B divided by the intensity measured at A.
2022-05-17