Show that the propagation of plane EM waves in an ionized gas leads to refractive index \( n = (1 - \frac{\omega_p^2}{\omega^2})^{1/2} \), where the symbols have their usual meaning.

Another Explanation (5): Starting from Maxwell's equations and assuming a plane wave solution in a plasma with electron density \(N\), the wave equation becomes: \( \nabla^2 \vec{E} - \frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2} = \mu_0 \frac{\partial \vec{J}}{\partial t} \) where \( \vec{J} = -e N \vec{v} \) is the current density, and \( \vec{v} \) is the electron velocity. Using the equation of motion for electrons in the plasma \( m_e \frac{\partial \vec{v}}{\partial t} = -e \vec{E} \) and assuming a harmonic time dependence \( e^{-i\omega t} \), we get: \( \vec{J} = \frac{e^2 N}{i m_e \omega} \vec{E} \) Substituting this into the wave equation and simplifying, leads to a modified wave equation with a modified permittivity: \( \nabla^2 \vec{E} + \frac{\omega^2}{c^2} \left( 1 - \frac{\omega_p^2}{\omega^2} \right) \vec{E} = 0 \) where \( \omega_p = \sqrt{\frac{e^2 N}{\epsilon_0 m_e}} \) is the plasma frequency. The refractive index \(n\) is defined as the ratio of the speed of light in vacuum to the speed of light in the medium. Comparing the above equation to the standard wave equation, we identify the refractive index as: \( n = \sqrt{1 - \frac{\omega_p^2}{\omega^2}} \)