Find the phase velocity of the wave (Plasma Physics)

[Firben]

Homework Statement

An electromagnetic wave with the frequency f = 1.4 GHz is propagating in the z-direction from vacuum in a plasma with the magnetic field B_0 = 0.1T z^. The plasma density is 1.0*10^17 particles/m^3. The wave is the separated into a R- and L-wave in the plasma.

Homework Equations

w_c = |q|*B/m cyclotron angular resonance
w_p = (n_0*e^2/(ε*m_e))^(1/2) plasma angular frequency
(ck/w)^2 = 1-(w_p^2*w^2)/(1-(w_c/w)) R-Wave (whistler)
(ck/w)^2 = 1-(w_p^2*w^2)/(1+(w_c/w)) L-wave

(v_ph)_R = (w/ck)_R = ((w(w-w_c))/(w^2-w*w_c-w_p^2))^(1/2)
(v_ph)_L = (w/ck)_R = ((w(w+w_c))/(w^2+w*w_c+w_p^2))^(1/2)

B_0 = 0.1T z^
f = 1.4 GHz = 8.8*10^9 rad/s
w_p = 1.78*10^10 rad/s
w_c = 1.76 * 10^10 rad/s

The Attempt at a Solution

Inserting these values into the formula for the phase velocity, i get the following

(v_ph)_R = c*0.443 = 1.329 * 10^8 m/s,
since the phase velocity is defined as w/k = v_ph. So i multiple it with c
correct value is v_ph = 6.6*10^6 m/s
(v_ph)_L = no wave, since it is imaginary

Then i want to know which polarization the wave has after z=1 m

 

[Dazed&Confused]

Can you write it with in LaTeX?

 

[Firben]

Homework Statement

An electromagnetic wave with the frequency f = 1.4 GHz is propagating in the z-direction from vacuum in a plasma with the magnetic field $B_0 = 0.1T \hat z$. The plasma density is $1.0*10^{17}$ $\frac {particles} {m^3}$. The wave is then separated into a R- and L-wave in the plasma.

2. Homework Equations
$w_c = \frac {|q|*B} m$ cyclotron angular resonance

$w_p = \sqrt \frac {n_0*e^2 } {ε*m_e}$ plasma angular frequency

$(\frac {ck} {w})^2 = 1- \frac {w_p^2/w^2} {1-(w_c/w)}$ R-Wave (whistler)
$(\frac {ck} {w})^2 = 1- \frac {w_p^2/w^2 } {1+(w_c/w)}$ L-wave

Phase Velocity:

$(v_{ph})_R = (\frac {w} {ck})_R = \sqrt \frac {w(w-w_c) } {w^2-w*w_c-w_p^2}$
$(v_{ph})_L = (\frac {w} {ck})_L = \sqrt \frac {w(w+w_c) } {w^2+w*w_c-w_p^2 }$

$B_0 = 0.1T \hat z$
$f = 1.4 GHz = 8.8*10^9$ rad/s
$w_p = 1.78*10^{10}$rad/s
$w_c = 1.76 * 10^{10}$ rad/s

3. The Attempt at a Solution

Inserting these values into the formula for the phase velocity, then i got the following

$(v_{ph})_R = c*0.443 = 1.329 * 10^8$m/s,
since the phase velocity is defined as $\frac {w} {k} = v_{ph}$. So i multiple it with c.
The correct value is $v_{ph} = 6.6*10^6$m/s
$(v_{ph})_L$ = no wave, since it is imaginary

How can i know which polarization the wave has after z=1 m ?

 

[Firben]

Why is my solution wrong ?

 

[Firben]

Im still stuck, why do i get a different answer ?