Natural and Laboratory Plasmas II – Example sheet 1
Orbit Theory and Radiation from a Charged Particle
May 2004 (some errors corrected)
DATA
Electron mass
Proton mass
Speed of light
Elementary charge
Boltzmann constant
Earth Mass
Earth Radius
me = 9.11  10-31 kg
mp = 1.673  10-27 kg
c = 2.998  108 ms-1
e = 1.602 10-19 C
kB = 1.381  10-23 J K-1
M = 5.794  1024 kg
R = 6.731  106 m
1. Calculate the Larmor radius, rL, of
- an electron travelling at 0.6c, with pitch angle 25o, in a coronal magnetic field of 10-3 T
- an alpha particle in a tokamak plasma with temperature 106K and B = 8T (here, calculate
the maximum value of rL)
- a 10keV electron in the Earth’s magnetic field at a distance of 5R above its surface (R is
the Earth’s radius). Assume that the equatorial field strength at the Earth’s surface is 3 
10-5 T.
2. Show that the gravitational force on a 1eV proton in the Earth’s ionosphere (which extends from
about 50km above the Earth to 10 R) is negligible compared to the Lorentz force it experiences. You
may use the value of the Earth’s magnetic field from question (1).
3. A particle with charge q is emitted from the origin with momentum p, directed at angle  to a
uniform magnetic field B which lies in the z-direction. At what point does the particle next intersect
the z-axis?
4. Calculate the loss-cone angle for charged particles at the midpoint of a symmetric magnetic bottle of
length L, with longitudinal magnetic field strength Bz which varying as Bz(z) = Bo(1+(z/L)2). How
does the loss-cone angle vary with position along the bottle?
5. Show that the magnetic field at the mirror point for a particle with speed v is
2
 v 

B m  B ( z )
v( z ) 
where v(z) is the perpendicular speed of the particle at position z, and B(z) is the z-component of the
magnetic field at this position. Hence, show that – in the field structure given above – the time, ,
taken for a trapped particle on the z-axis to bounce between its two mirror points at z = L and z = -L
is
 
2L
v
6. A column of electrons has density ne = 1010m-3, and radius a = 103 cm. It is confined by a 0.1T
magnetic field, with B in the positive z-direction. Use Poisson’s equation to work out the electrostatic
field due to the electrons, and hence the magnitude and direction of the EB drift at r = a.
If the electrons are flowing along the column at uniform speed, how will the EB drift change?
7. Suppose that a region of free space has a magnetic field B  B ( r )eˆ  , where  is the azimuthal
direction. Show that the guiding centre of a gyrating charged particle of mass m and charge q has a
combination of curvature and gradient drifts given by
v gc  
m RC  B  2 1 2 
v ||  v  
q RC2 B 2 
2 
where Rc is the radius of curvature of the field lines, and v||, v are the parallel and perpendicular
components of the particle velocity.
8. Show for a relativistic accelerating particle, with collinear velocity and acceleration vectors that the
angle to these vectors at which the peak power is radiated is given by
cos  max 

1
1  15 2
3

1/ 2

1
where  = v/c
9. Prove that
d
dt'
where R̂ is a unit vector.

 Rˆ  Rˆ  



  1 Rˆ  Rˆ     



2