A plane electromagnetic wave `E_(z) = 100 "cos" (6xx10^(8)t +4x)` V/m propagating in medium of dielectric constant is

Options

A. 1.5
B. 2
C. 2.4
D. 4

Answer

To solve the problem, we need to determine the dielectric constant of the medium in which the given electromagnetic wave is propagating. The wave is described by the equation:

\[ E_z = 100 \cos(6 \times 10^8 t + 4x) \, \text{V/m} \]

### Step-by-Step Solution:

1. **Identify the parameters from the wave equation**:
The given wave equation can be compared to the standard form of an electromagnetic wave:
\[ E = E_0 \cos(\omega t + kx) \]
From the equation \( E_z = 100 \cos(6 \times 10^8 t + 4x) \), we can identify:
- \( E_0 = 100 \, \text{V/m} \)
- \( \omega = 6 \times 10^8 \, \text{rad/s} \)
- \( k = 4 \, \text{rad/m} \)

2. **Calculate the speed of light in the medium**:
The relationship between the wave number \( k \), angular frequency \( \omega \), and the speed of light \( c \) in the medium is given by:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{6 \times 10^8}{4} = 1.5 \times 10^8 \, \text{m/s} \]

3. **Relate the speed of light in the medium to the dielectric constant**:
The speed of light in a medium is related to the speed of light in vacuum \( c \) and the dielectric constant \( \epsilon_r \) (or refractive index \( n \)) by the formula:
\[ v = \frac{c}{n} \]
where \( c \approx 3 \times 10^8 \, \text{m/s} \). Rearranging gives:
\[ n = \frac{c}{v} \]

4. **Calculate the dielectric constant**:
Substituting the values:
\[ n = \frac{3 \times 10^8}{1.5 \times 10^8} = 2 \]

5. **Conclusion**:
The dielectric constant of the medium is \( n = 2 \).

### Final Answer:
The dielectric constant of the medium is **2**.