## Thursday, December 9, 2021

### Solutions To Radio Astronomy Plasma Physics Problems

1) Substitute  the values for   vx , v y  and  vz    earlier obtained into equations (1),  (2) and (3) giving Maxwell's  curl H equation in terms of rectangular components to get the 3 resulting plasma frequency equations. Then show if the frequency is much greater than the gyrofrequency,  i.e.

w  >>  g

All the equations become identical in form

Solution:   The equations are:

1)  (Ñ x H) x   Ne vx  +   εo Ex

2)  (Ñ x H) y  =  Ne v y  +   εo Ey

3)  (Ñ x Hz  =  N e vz  +   εo Ez

And we know:

vx = { j  (m /q) E x  E y B z}/ B -  m 2 2 /q 2

vy = { j  (m /q) E y  x B z}/ B -  m 2 2 /q 2

vz = - j q E /m w

Substitute the v's into the correct equations (1), (2) and (3), and let q = e  ( 1.6 x  10-19 C ) to get:

1b)  (Ñ x H) x
Ne {j  (m /q) E x  E y B z}/ B -  m 2 2 /e 2  +     εo Ex

2b)     (Ñ x H) y
Ne { j  (m /q) E y  x B z}/ B -  m 2 2 /e }    +   εo Ey

3b)   x Hz  =  Ne (- j e E m w)   +   εo Ez

After some algebra, we obtain:

x H) x
w εo  x   [1 + N2  /εom ( w g 2     w 2)]

NE q   w  /εom ( w g 2     w 2

x H) y

w εo  y   [1 + N2 /εom ( w g 2     w 2)]

-    Nq w g  / m( w g 2     w 2

x Hz  =

w εo  z   [1 -  N2 /εom w 2]

Now, assume:
w  >>  g

Then we see:

x H) x
w εo  x   [1 + N2 /εom ( w 2)]

x H) y

w εo  y   [1 + N2 /εom ( w 2)]

x Hz  =

w εo  E z   [1 -  N e 2 /εom w 2]

I.e.  All the relations become identical in form.

2) The Earth's ionosphere has a plasma number density of
10 12     electrons per cubic meter.  Find the (critical)  radio frequency of this source.

Solution:

The critical frequency is given by:

o     =   9 Ö N

Where N  = 10 12

Then:  f o     =   9 Ö 10 12

=   9  x 10 6   cycles per second

Or:   9 MHz