Trouble with Plasma Frequency Calculation

[Blanchdog]

Homework Statement

What is the complex refractive index of the ionosphere for an AM radio station at 1160 KHz? Is this frequency above or below the plasma frequency? Assume an electron density of 1974 electrons per cubic meter.

Relevant Equations

The Plasma frequency is given by $\omega_p^2=\frac{N q_e }{\epsilon_0 m_e}$
The complex index of refraction is given by $\mathcal{N}^2 = (n + \text i\kappa)^2 = 1+ \frac{\omega_p^2}{\omega_0^2-\text i \omega \gamma - \omega^2}$ However, since we are dealing with a plasma, $\omega_0 = 0$ and $\gamma = 0$ and so $\mathcal{N}^2 = 1- \frac{\omega_p^2}{ \omega^2}$

This would appear to be a straightforward algebra problem, but it just doesn't pass the smell test for me. The issue might be with the number of electrons per cubic meter, as that was calculated in a previous problem, so let me know if that number seems wrong.

My plasma frequency ($\nu$) not ($\omega$) came out to 399 Hz, which is nowhere near 1160 KHz. I calculated my index of refraction to be almost arbitrarily close to 1 with no imaginary part, which seems weird since the problem asked for a complex index of refraction.

 

[TSny]

The issue might be with the number of electrons per cubic meter, as that was calculated in a previous problem, so let me know if that number seems wrong.

Your value of 1974 m^-3 for the electron density seems to be way too small. I think ionosphere electron densities are on the order of 10¹⁰ m^-3.

 

[TSny]

The Plasma frequency is given by $\omega_p^2=\frac{N q_e }{\epsilon_0 m_e}$

$q_e$ should be squared.

 

[Blanchdog]

$q_e$ should be squared.

I think that's just a typo since it doesn't look like I forgot to square it in my written work.

I calculated the number of available free electrons as follows, then was asked to assume the same number of free electrons:

The index of refraction of the ionosphere is $\mathcal{N} = 0.9$ for an FM station at $\nu =\frac{\omega}{2\pi} = 100$MHz. Assume $\omega_0 = \gamma = 0$.

Then
$$\mathcal{N}^2 = 1+ \frac{\omega_p^2}{\omega_0^2-\text{i}\gamma\omega -\omega^2}=1-\frac{\omega_p^2}{\omega^2} = 1- \frac{Nq_e^2}{\epsilon_0 m_e \omega^2}$$
Rearranging, we get
$$N = \frac{(1-\mathcal{N}^2)\omega^2\epsilon_0 m_e}{q_e^2}$$

Aaaaand I just found my error. When I did this on paper I forgot to square $\omega$, decreasing my electrons per cubic meter by 8 orders of magnitude or so. Thanks for the estimate on ionosphere densities, I might never have known something was amiss otherwise.

 

[hutchphd]

For @Blanchdog
Given that you said you know
$\mathcal{N}^2 = 1- \frac{\omega_p^2}{ \omega^2}$
And that The index of refraction of the ionosphere is N=0.9 for an FM station at ν=ω2π=100MHz.

Why didn't you just change the frequency $\omega$ by 1.16/100 ?

edited by the lets-be-nice police to save this post.

 

[Blanchdog]

For @Blanchdog
Given that you said you know
$\mathcal{N}^2 = 1- \frac{\omega_p^2}{ \omega^2}$
And that The index of refraction of the ionosphere is N=0.9 for an FM station at ν=ω2π=100MHz.

Why didn't you just change the frequency $\omega$ by 1.16/100 ?

1) Your calculation of $\nu$ is wrong.
2) The arithmetic of that part of the problem wasn't even part of the question because it's so trivial

Edits were done by mentor: (the lets-be-nice police have interceded here to save this post)