Plasma Frequency

The plasma frequency is the natural oscillation frequency of electrons in a plasma. If electrons are displaced from their equilibrium positions, the restoring electric force causes them to oscillate at the plasma frequency.

Electron Plasma Frequency

$\omega_{pe} = \sqrt{\frac{n e^2}{\varepsilon_0 m_e}}$

In cycles per second (Hz):

$f_{pe} = \frac{\omega_{pe}}{2\pi}$

Where:

• $n$ = electron number density (m⁻³)
• $e = 1.602 \times 10^{-19}$ C
• $\varepsilon_0 = 8.854 \times 10^{-12}$ F/m
• $m_e = 9.109 \times 10^{-31}$ kg

Ion Plasma Frequency

Ions also have a plasma frequency, but much lower due to their larger mass:

$\omega_{pi} = \sqrt{\frac{n e^2}{\varepsilon_0 m_i}}$

where $m_i = A \times 1.673 \times 10^{-27}$ kg, and $A$ is the atomic mass number (e.g., $A=1$ for hydrogen/protons).

Physical Significance

• Electromagnetic waves with frequency $f < f_{pe}$ cannot propagate through the plasma — they are reflected. This is why the ionosphere reflects AM radio waves!
• The ratio $\omega_{pe} / \omega_{pi} = \sqrt{m_i / m_e} \approx 43$ for hydrogen — electrons oscillate much faster than ions.

Implement the three functions below to compute these frequencies.