Estimation of Plasma Reflection Wavelength
We are given:
- Electron number density, \( N \approx 4 \times 10^{27} \, \text{m}^{-3} \)
- Permittivity of free space, \( \varepsilon_0 = 10^{-11} \, \text{F/m} \)
- Electron mass, \( m = 10^{-30} \, \text{kg} \)
- Elementary charge, \( e = 1.6 \times 10^{-19} \, \text{C} \)
1. Formula for Plasma Frequency
\( \omega_p = \sqrt{\frac{N e^2}{\varepsilon_0 m}} \)
2. Substitute the Values
\( \omega_p = \sqrt{ \frac{(4 \times 10^{27}) \times (1.6 \times 10^{-19})^2 }{ (10^{-11}) \times (10^{-30}) } } \)
Calculate step-by-step:
- \( e^2 = (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \)
- Numerator: \( 4 \times 10^{27} \times 2.56 \times 10^{-38} = 1.024 \times 10^{-10} \)
- Denominator: \( 10^{-11} \times 10^{-30} = 10^{-41} \)
- So, \( \omega_p = \sqrt{ \frac{1.024 \times 10^{-10} }{10^{-41}} } = \sqrt{1.024 \times 10^{31}} \)
- \( \omega_p \approx 1.012 \times 10^{15} \, \text{rad/s} \)
3. Convert to Wavelength
\( \lambda_p = \frac{2\pi c}{\omega_p} \), where \( c = 3 \times 10^8 \, \text{m/s} \)
- \( \lambda_p = \frac{2 \pi \times 3 \times 10^8}{1.012 \times 10^{15}} \)
- \( \lambda_p \approx \frac{1.884 \times 10^9}{1.012 \times 10^{15}} \approx 1.86 \times 10^{-6} \, \text{m} \)
🔍 The estimated wavelength for plasma reflection is:
\( \lambda_p \approx 1.86 \, \mu\text{m} \) (micrometers)
