## Thursday, December 23, 2021

### Solutions To Radio Astronomy Plasma Physics Problems (2)

1)  A space plasma with electron number density  10 12 /cm 3  features a magnetic field of 0.0001 T (Tesla).  Find: a) the electron plasma frequency, b) the cyclotron frequency and c) the hybrid frequency.

Thence or otherwise find the refractive index of the plasma and whether ordinary or extraordinary wave propagation can be expected.

Solns.

a) The electron plasma frequency:

w e     =  [ne e2/ me  εo½

Where:  ne  10 12  /cm 3  10 18  /m 3

e  =  1.6 x 10-19  C

me   =  9.1 x 10-31  kg

ε o  =  8.85 x 10-12  F/m

Then:

w e

[10 18 m-3 (1.6 x 10-19  C2  /9.1 x 10-31  kg (8.85 x 10-12  F/m)½

w e  = 5.63 x 10 10   c/s

b) The cyclotron frequency:

c   =  qB/ m e

Where:  q  =  e  =  1.6 x 10-19  C

B = 0.0001 T

me   =  9.1 x 10-31  kg

Then:  c  (1.6 x 10-19  C 0.0001 T /9.1 x 10-31  kg

=  1.76 x 10 7   c/s

c) The hybrid frequency

w H 2 =    w e 2 +   c 2

Or:   w H    =   [  w e 2 +   c 2 ½

w H    =   [  (5.63 x 10 10   c/s) 2 +  (1.76 x 10 7   c/s) 2 ½

w H    =      5.63 x 10 10   c/s

The refractive index of the plasma can only apply to the ordinary wave since:

For the extra ordinary wave: n x 2 =   1   -   X /  ( -  Y)

But:   X  =   w e 2 w 2     and   Y  =  w H 2 w 2

However, on computation we find:

w e    »    w,    w H   »    w  =  5.63 x 10 10   c/s

So:  X  =  1      and   Y  =  1   whence:

n x 2 =   1   -   X /  ( -  Y)  =    1 -   1/ (1 -  1)  = 1 - 1/0

So:  n x   =  [1 -  ¥] ½       So  indeterminate

But for the ordinary wave:

n o 2 =   1   -   X /  ( 1  Y)

So:  n o   =  [1 -  1/ 1 + 1] ½   =  [1 -  1/ 2] ½

n o   =  [1/ 2] ½

n o   =  Ö 2 / 2 =   0.707

2)  For the plasma in (1)  and assuming condition (3) (f  > o )  find the amount of rotation of the plane of linear polarization expected, assuming a plasma thickness  dz =  0.15 m.

Soln.  The primary goal here is to render all units consistent.

If we use the formula given for the magnitude of rotation in radians, i.e.:

2.38  x  10    [ 1  f 2   ò n e  H  dz ]

Then all the units must be c.g.s.

So:  ne  10 12 cm-3

H =    (0.0001 T) (10,000 G/T) =  1 Gauss

=    Ö w2 p

w      =  [ne e2/ me  εo½

where: e  =   4.8  10  -10   esu (electrostatic units)

me   9.1 x 10-28   g

εo    =  1/ 4 p

Then after doing the computation we obtain:

=    Ö  5.6  10  10   c/s  / 2 p  =   8.9  10    c/s

Then compute:

2.38  x  10    [ 1  f 2   ò n e  H  dz ]

=  2.38  x  10 [ 1  (8.9  10    c/s) 2  10 12 cm-3 (1)  15 cm]

The  magnitude of the rotation of the linear plane of polarization is:  0.44 rad, or about 25.2 deg.