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UNIVERSITY COLLEGE LONDON
EXAMINATION FOR INTERNAL STUDENTS
MODULE CODE
ASSESSMENT
PATTERN
MODULE NAME
PHASM465
PHASM465A
Space Plasma and Magnetospheric Physics
(Masters Level)
DATE
TIME
Monday 30 April 2018
10:00
TIME ALLOWED hrs 30 mins
This paper is suitable for candidates who attended classes for this
module in the following academic year(s):
Year
Suitable for all candidates
EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM
PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE
2016/17-PHASM465A-001-EXAM-Physics and Astronomy 14
2016 University College London
TURN OVER
Answer THREE questions.
Note: only three questions will be marked
The numbers in square brackets in the right-hand margin indicate provisional allocation of
maximum possible marks for different parts of each question.
Vectors are denoted by bold-faced type, e.g., B, while scalar quantities, including the
magnitude of the corresponding vector, are in italic type, e.g. and |B|.
The following may be assumed, if required:
Speed of light in vacuo 2.998 108 s”1
Universal gravitational constant 6.674 10″” m2 kg’2
Boltzmann’s constant its 1.381 10’23 J K”1
Proton mass m, 1.673 10’27 kg
Electron mass me 9.109 10’31 kg
Electron charge 1.602 1O’I9C
Electron volt unit eV 1.602 1019
Earth’s Radius RE 6.378 106
Equatorial Magnetic field at Earth’s surface Be 3.1 10’5
Permittivity of free space eo 8.85 10″12 m”1
Permeability of free space no 4rc 10″7 m’1
MHD equations:
v (7 where mass density and number
—+ V-(/ v) A—hv-V -VP+jxB density are related byp nm and
3/ & (mt’ me), is the fluid flow velocity,
|_ v.V|(pp”r) -vxB+77J is the pressure tensor, is the specific
vd’ heat ratio and tj is resistivity.
The strength of the Earth’s magnetic field at (L, A), in the dipole field approximation, is:
where Be is the dipole magnetic field strength at the Earth’s surface at the equator, is the
Mcllwain parameter (radial distance to where the dipole magnetic field line crosses the
magnetic equator in units of Earth radii).
PHASM465/2018 CONTINUED
Question 1.
a) Draw large, clearly labelled diagram and explain the motion of protons in vacuum in
which there is uniform magnetic field and no electric field. Illustrate and explain the
cases in which the proton is initially moving with velocity vector which is:
i) Parallel to the magnetic field vector;
ii) At 45° to the magnetic field vector;
iii) At 90° to the magnetic field vector.
Briefly explain the differences expected in the above motions if the charged particle
considered were an electron, rather than a proton.
[4 Marks]
b) Consider the general case of charged particle with velocity (vx, vy, vz) in a region with
uniform, time-stationary magnetic field described by the vector (Bx, 0, 0) and
uniform, steady electric field vector (0, 0, Ez). Write down an expression for the
Lorentz force and use it to demonstrate that the particle motion consists of components:
i) Steady motion parallel to the magnetic field vector;
ii) Circular motion in the plane perpendicular to the magnetic field vector;
iii) steady drift motion in the direction.
[4 Marks]
c) Draw large, clearly labelled diagram to illustrate the motion of both ions and electrons in
the electromagnetic fields described in part (b), in the case that the particle velocity is
initially perpendicular to the magnetic field. Write down expressions for the radius and
frequency of the circular motion and the speed of the steady drift motion.
[4 Marks]
d) The general expressions for the electric field drift velocity vE and the magnetic field drift
velocity vM in dipole magnetic field are given by:
ExB BxVB
vE -gT- (W 2W)
where W± and W\ are the perpendicular and parallel particle energy respectively.
Consider point at the magnetic equator on field line of Mcllwain parameter L. Show
that the ratio of the magnetic field drift to the electric field drift, at this point, is:
vE
[4 Marks]
e) Consider singly charged particle at the magnetic equator, on magnetic field line with
6, and moving with velocity vector perpendicular to the magnetic field. If the combined
convection and corotation electric field strength is 0.4 mV m”‘ at 6, what is the particle
energy, in eV, for which the magnetic and electric field drifts are of equal magnitude
Comment briefly on the relative impact of this for ions and electrons of this energy.
[4 Marks]
PHASM465/2018 CONTINUED
Question
a) What are the types of periodic motion that may be performed by charged particles
trapped within planetary dipole magnetic field Briefly describe the nature and
physical origin of each type of motion and clearly illustrate your answer with
diagram.
[4 Marks]
b) Each of the types of periodic motion is associated with an adiabatic invariant. For
each invariant, give the name of the invariant associated with each of the motions.
State the physical quantity associated with each invariant that is conserved during each
motion.
[4 Marks]
c) For each of the adiabatic invariants discussed above, briefly describe physical
circumstances which may arise in the magnetosphere and lead to violation of the
invariant. Describe also the consequences for the particle motion in each case of
violation of the invariant.
[4 Marks]
d) Consider the exchange of kinetic energy associated with the components of motion of
charged particle in the directions perpendicular and parallel to the magnetic field in
order to demonstrate mathematically that the first adiabatic invariant, //, is conserved
in time-stationary but non-uniform magnetic field.
[4 Marks]
e) In dipole magnetic field, the drift velocity of charged particle of mass m, equatorial
pitch angle Okq and total kinetic energy W, is approximately given by:
3L2W
where is the Mcllwain parameter that characterises the magnetic field at the
location of the particle.
population of electrons with an equatorial pitch angle of 90° and energy of MeV
is observed by spacecraft located at geocentric distance of Re. The location of
the magnetopause is disturbed by changes in the solar wind dynamic pressure which
launches ULF waves into the magnetosphere. What frequencies of ULF waves are
likely to disturb the stability of the observed electron population
[4 Marks]
PHASM465/2018 CONTINUED
Question 3.
a) Describe the conditions under which fast mode shocks may form within the
heliosphere. Identify and describe examples of where they may be found.
[4 Marks]
b) Use the MHD and Maxwell’s equations to derive the following Rankine-Hugoniot
equations, valid at steady-state, 1-dimensional collisionless plasma shock:
where ux, Bx are the components of the plasma velocity and magnetic field normal to
the shock surface, and is the plasma density.
Explain the physical relevance of these equations to the changes in the field and plasma
as they pass through fast-mode collisionless shock.
[4 Marks]
c) How does the magnetic field direction affect the structure of fast mode shock What
mechanisms are responsible for particle acceleration to very high energies at or near the
bow shock surface, and how do these depend on the field direction
[4 Marks]
d) The rotation of the Sun causes the interplanetary magnetic field to become wound up into
the Parker spiral configuration, since the frame co-rotating with the Sun has an additional
azimuthal component u# -Q r. In a spherically symmetric model, the frozen-in flux
condition implies that
Br ur ur
Use Maxwell’s equations to determine the variation of the radial component of the
magnetic field, Br, with radial distance r, and thus use the above condition to determine
the azimuthal component, 5