A dence collection of equal number of electrona and positive ions is called netural plasma. Certain solids contianing fixed positive ions surroundedby free electrons can be treated as neytral plasma. Let 'N' be the numbrer density of free electrons, each of mass '`m`'. When the elctrons are subjected to an eletric field, they are displaced relatively away from the heavy positive ions. if the electric field becomes zero, the electrons begin to oscillate about the positive ions with a natural angular frequency '`omega_(P)'` which is called the plasma frequency. to sustain the oscillations, a time varying electric field needs to be applied that has an angular frequrncy `omega`, where a part of the energy is absorbed and a part of it is reflected. As `omega` approaches `omega_(p)` all the free electrons are set to resonance together and all the energy is reflected. this is the explaination of high reflectivity of metals.
(2) Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons `N~~ 4 xx 10^(27)m^(-3)`. Taking `epsilon_(0) = 10^(11)` and mass `m~~ 10^(-30)`, where these quantities are in proper `SI` units.

Options

A. `800 nm`
B. `600 nm`
C. `300 nm`
D. `200 nm`

Explanation

To estimate the wavelength at which plasma reflection will occur for a metal with an electron density \( N = 4 \times 10^{27} \, \text{m}^{-3} \), we will follow these steps:

### Step 1: Understand the Plasma Frequency Formula
The plasma frequency \( \omega_p \) is given by the formula:
\[ \omega_p = \sqrt{\frac{N e^2}{m \epsilon_0}} \]
where:
- \( N \) is the number density of free electrons,
- \( e \) is the charge of an electron (\( e \approx 1.6 \times 10^{-19} \, \text{C} \)),
- \( m \) is the mass of an electron (\( m \approx 10^{-30} \, \text{kg} \)),
- \( \epsilon_0 \) is the permittivity of free space (\( \epsilon_0 \approx 8.85 \times 10^{-12} \, \text{F/m} \)).

### Step 2: Relate Angular Frequency to Wavelength
The angular frequency \( \omega \) can be related to the frequency \( f \) and wavelength \( \lambda \) as follows:
\[ \omega = 2 \pi f = \frac{2 \pi c}{\lambda} \]
where \( c \) is the speed of light (\( c \approx 3 \times 10^8 \, \text{m/s} \)).

### Step 3: Set Up the Equation
At resonance, we set \( \omega = \omega_p \):
\[ \frac{2 \pi c}{\lambda} = \sqrt{\frac{N e^2}{m \epsilon_0}} \]

### Step 4: Solve for Wavelength \( \lambda \)
Rearranging the equation to solve for \( \lambda \):
\[ \lambda = \frac{2 \pi c}{\sqrt{\frac{N e^2}{m \epsilon_0}}} \]

### Step 5: Substitute the Known Values
Now, substitute the known values into the equation:
- \( N = 4 \times 10^{27} \, \text{m}^{-3} \)
- \( e = 1.6 \times 10^{-19} \, \text{C} \)
- \( m = 10^{-30} \, \text{kg} \)
- \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \)
- \( c = 3 \times 10^8 \, \text{m/s} \)

Calculating \( \lambda \):
\[ \lambda = \frac{2 \pi (3 \times 10^8)}{\sqrt{\frac{(4 \times 10^{27})(1.6 \times 10^{-19})^2}{(10^{-30})(8.85 \times 10^{-12})}}} \]

### Step 6: Calculate the Denominator
Calculating the denominator:
\[ \frac{(4 \times 10^{27})(1.6 \times 10^{-19})^2}{(10^{-30})(8.85 \times 10^{-12})} = \frac{(4 \times 10^{27})(2.56 \times 10^{-38})}{(10^{-30})(8.85 \times 10^{-12})} \]
\[ = \frac{10.24 \times 10^{-11}}{8.85 \times 10^{-12}} \approx 1.157 \times 10^{1} \approx 11.57 \]

### Step 7: Calculate \( \sqrt{11.57} \)
\[ \sqrt{11.57} \approx 3.4 \]

### Step 8: Final Calculation of Wavelength
Now substituting back:
\[ \lambda = \frac{2 \pi (3 \times 10^8)}{3.4} \approx \frac{6.2832 \times 3 \times 10^8}{3.4} \approx \frac{1.88496 \times 10^9}{3.4} \approx 554.4 \, \text{nm} \]

### Conclusion
The estimated wavelength at which plasma reflection will occur is approximately \( 554.4 \, \text{nm} \).