An EM wave from air enters a medium. The electric fields are
`overset(vec)E_(1)=E_(01)hatx cos[2piv((z)/(c )-t)]` in air and
`overset(vec)E=E_(02)hatx cos [k(2z-ct)]` in medium, where the wave number k and frequency v refer to their values in air. The medium is non-magnetic. If `in_(r_(1))` and `in_(r_(2))` refer to relative permittivities of air and medium respectively, which of the following options is correct ?

Options

A. `(in_(r_(1)))/(in_(r_(1)))=(1)/(2)`
B. `(in_(r_(1)))/(in_(r_(1)))=4`
C. `(in_(r_(1)))/(in_(r_(1)))=2`
D. `(in_(r_(1)))/(in_(r_(1)))=(1)/(4)`

Answer

To solve the problem, we need to analyze the given electric fields of the electromagnetic waves in both air and the medium, and then relate their speeds to the relative permittivities.

### Step-by-Step Solution:

1. **Identify the Electric Field Equations**:
- In air:
\[ \vec{E_1} = E_{01} \hat{x} \cos\left(2\pi v \left(\frac{z}{c} - t\right)\right) \]
- In the medium:
\[ \vec{E_2} = E_{02} \hat{x} \cos\left(k(2z - ct)\right) \]

2. **Rewrite the Electric Field in Standard Form**:
- The standard form of the electric field is:
\[ E = E_0 \cos(kz - \omega t) \]
- For the air wave, we can rewrite it as:
\[ \vec{E_1} = E_{01} \hat{x} \cos\left(2\pi v z - 2\pi vt\right) \]
- For the medium, we rewrite it as:
\[ \vec{E_2} = E_{02} \hat{x} \cos\left(2kz - ckt\right) \]

3. **Determine the Angular Frequency and Wave Number**:
- For air:
- Angular frequency \( \omega_1 = 2\pi v \)
- Wave number \( k_1 = 2\pi v/c \)
- For the medium:
- Angular frequency \( \omega_2 = ck \)
- Wave number \( k_2 = 2k \)

4. **Calculate the Speed of the Waves**:
- Speed in air:
\[ V_1 = \frac{\omega_1}{k_1} = \frac{2\pi v}{2\pi v/c} = c \]
- Speed in the medium:
\[ V_2 = \frac{\omega_2}{k_2} = \frac{ck}{2k} = \frac{c}{2} \]

5. **Relate Speeds to Permittivities**:
- The speed of an electromagnetic wave in a medium is given by:
\[ V = \frac{1}{\sqrt{\mu_0 \epsilon_r}} \]
- For air:
\[ V_1 = c = \frac{1}{\sqrt{\mu_0 \epsilon_{r1}}} \]
- For the medium:
\[ V_2 = \frac{c}{2} = \frac{1}{\sqrt{\mu_0 \epsilon_{r2}}} \]

6. **Set Up the Ratio of Speeds**:
- From the equations of speed:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_{r1}}} \]
\[ \frac{c}{2} = \frac{1}{\sqrt{\mu_0 \epsilon_{r2}}} \]
- Squaring both sides and rearranging gives:
\[ \epsilon_{r1} = \mu_0 c^2 \]
\[ \epsilon_{r2} = 4\mu_0 c^2 \]

7. **Find the Ratio of Relative Permittivities**:
- Thus, we have:
\[ \frac{\epsilon_{r2}}{\epsilon_{r1}} = 4 \]

### Conclusion:
The correct relationship between the relative permittivities is:
\[ \frac{\epsilon_{r2}}{\epsilon_{r1}} = 4 \]