Estimate the wavelength at which plasma reflection will occur for metal having the density of electrons `N ~~4xx10^(27)m^(-3)` . Take `epsilon_0=10^(11) and m = 10^(-30)` , where these quantities are in proper SI units.

Options

A. 800 nm
B. 600 nm
C. 400 nm
D. 200 nm

Explanation

To estimate the wavelength at which plasma reflection will occur for a metal with a given electron density, we can follow these steps:

### Step 1: Understand the Plasma Frequency
The plasma frequency (\( \nu_p \)) is the frequency at which electrons in a plasma oscillate. It is given by the formula:
\[ \nu_p = \frac{1}{2\pi} \sqrt{\frac{n e^2}{m \epsilon_0}} \]
where:
- \( n \) is the density of electrons (in \( m^{-3} \)),
- \( e \) is the charge of an electron (\( 1.6 \times 10^{-19} \, C \)),
- \( m \) is the mass of an electron (\( 9.11 \times 10^{-31} \, kg \)),
- \( \epsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \, F/m \)).

### Step 2: Substitute the Values
Given:
- \( n = 4 \times 10^{27} \, m^{-3} \)
- \( e = 1.6 \times 10^{-19} \, C \)
- \( m = 10^{-30} \, kg \) (as per the question)
- \( \epsilon_0 = 10^{-11} \, F/m \) (as per the question)

Substituting these values into the formula for plasma frequency:
\[ \nu_p = \frac{1}{2\pi} \sqrt{\frac{(4 \times 10^{27}) (1.6 \times 10^{-19})^2}{(10^{-30})(10^{-11})}} \]

### Step 3: Calculate the Plasma Frequency
Calculating the numerator:
\[ (1.6 \times 10^{-19})^2 = 2.56 \times 10^{-38} \]
Now substituting this back:
\[ \nu_p = \frac{1}{2\pi} \sqrt{\frac{(4 \times 10^{27})(2.56 \times 10^{-38})}{(10^{-30})(10^{-11})}} \]
Calculating the denominator:
\[ (10^{-30})(10^{-11}) = 10^{-41} \]
Now substituting:
\[ \nu_p = \frac{1}{2\pi} \sqrt{\frac{(4 \times 10^{27})(2.56 \times 10^{-38})}{10^{-41}}} \]
\[ = \frac{1}{2\pi} \sqrt{(4 \times 2.56) \times 10^{27 + 41 - 38}} = \frac{1}{2\pi} \sqrt{10.24 \times 10^{30}} = \frac{1}{2\pi} \sqrt{10.24} \times 10^{15} \]
Calculating \( \sqrt{10.24} \approx 3.2 \):
\[ \nu_p \approx \frac{3.2 \times 10^{15}}{2\pi} \approx \frac{3.2 \times 10^{15}}{6.28} \approx 5.1 \times 10^{14} \, Hz \]

### Step 4: Calculate the Wavelength
The wavelength (\( \lambda \)) corresponding to the plasma frequency can be calculated using the formula:
\[ \lambda = \frac{c}{\nu_p} \]
where \( c \) is the speed of light (\( 3 \times 10^8 \, m/s \)):
\[ \lambda = \frac{3 \times 10^8}{5.1 \times 10^{14}} \approx 5.88 \times 10^{-7} \, m \]
This can be converted to nanometers:
\[ \lambda \approx 588 \, nm \]

### Final Answer
The estimated wavelength at which plasma reflection will occur is approximately \( 588 \, nm \).

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