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.
(1) Taking the electronic charge as 'e' and the permittivity as `'epsilon_(0)`'. use dimensional analysis to determine the correct expression for `omega_(p)`.

Options

A. `sqrt((Ne)/(mepsilon_(0)))`
B. `sqrt((mepsilon_(0))/(Ne))`
C. `sqrt((Ne^(2))/(mepsilon_(0)))`
D. `sqrt((mepsilon_(0))/(Ne^(2)))`

Explanation

To determine the plasma frequency \( \omega_p \) using dimensional analysis, we need to consider the relevant quantities: the charge of the electron \( e \), the permittivity of free space \( \epsilon_0 \), and the number density of free electrons \( N \) (which has dimensions of \( L^{-3} \)). The mass of the electron \( m \) is also relevant.

### Step-by-step Solution:

1. **Identify the Dimensions of Each Quantity**:
- Charge \( e \): The dimension of charge can be expressed in terms of current (ampere) and time. Thus, the dimension of charge is \( [e] = [I][T] = [A][T] \).
- Permittivity \( \epsilon_0 \): The dimension of permittivity can be derived from Coulomb's law. It is given as \( [\epsilon_0] = [M^{-1} L^{-3} T^4 A^2] \).
- Number density \( N \): This is the number of particles per unit volume, so its dimension is \( [N] = [L^{-3}] \).
- Mass \( m \): The dimension of mass is simply \( [M] \).
- Angular frequency \( \omega \): The dimension of angular frequency is \( [\omega] = [T^{-1}] \).

2. **Construct a Dimensional Formula for \( \omega_p \)**:
We want to find a combination of \( e \), \( \epsilon_0 \), \( N \), and \( m \) that results in the dimension of \( [T^{-1}] \).

Let’s assume:
\[ \omega_p \propto \sqrt{\frac{N e^2}{m \epsilon_0}} \]

3. **Determine the Dimensions of the Right Side**:
- The dimension of \( N e^2 \):
\[ [N e^2] = [L^{-3}] \cdot [A^2 T^2] = [L^{-3} A^2 T^2] \]
- The dimension of \( m \epsilon_0 \):
\[ [m \epsilon_0] = [M] \cdot [M^{-1} L^{-3} T^4 A^2] = [L^{-3} T^4 A^2] \]

4. **Combine the Dimensions**:
\[ \frac{N e^2}{m \epsilon_0} = \frac{[L^{-3} A^2 T^2]}{[L^{-3} T^4 A^2]} = [T^{-2}] \]

5. **Take the Square Root**:
\[ \sqrt{\frac{N e^2}{m \epsilon_0}} = [T^{-1}] \]

6. **Final Expression for Plasma Frequency**:
Thus, the plasma frequency \( \omega_p \) can be expressed as:
\[ \omega_p = \sqrt{\frac{N e^2}{m \epsilon_0}} \]

### Conclusion:
The correct expression for the plasma frequency \( \omega_p \) is:
\[ \omega_p = \sqrt{\frac{N e^2}{m \epsilon_0}} \]