Electromagnetic wave in a medium of `mu_(r) = 1` is given as `E = 20 sin (20 x 10^10 - 20 x 10^2 x)` . Dielectric constant of medium is `1/x` then find x

Answer

To solve the problem, we need to analyze the given electromagnetic wave equation and the relationship between the parameters involved.

### Given:
- The electromagnetic wave is represented as:
\[ E = 20 \sin(20 \times 10^{10} t - 20 \times 10^2 x) \]
- The relative permeability of the medium is given as:
\[ \mu_r = 1 \]
- The dielectric constant of the medium is given as:
\[ \epsilon_r = \frac{1}{x} \]

### Step 1: Identify the angular frequency (ω) and the wave number (k)
From the wave equation, we can identify:
- Angular frequency \( \omega = 20 \times 10^{10} \, \text{rad/s} \)
- Wave number \( k = 20 \times 10^2 \, \text{rad/m} \)

### Step 2: Calculate the phase velocity (v)
The phase velocity \( v \) of an electromagnetic wave is given by the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values:
\[ v = \frac{20 \times 10^{10}}{20 \times 10^2} \]
The \( 20 \) cancels out:
\[ v = \frac{10^{10}}{10^2} = 10^{8} \, \text{m/s} \]

### Step 3: Relate phase velocity to the medium's properties
The phase velocity \( v \) in a medium can also be expressed in terms of the speed of light \( c \) and the relative permittivity \( \epsilon_r \) and relative permeability \( \mu_r \):
\[ v = \frac{c}{\sqrt{\mu_r \epsilon_r}} \]
Given that \( c = 3 \times 10^8 \, \text{m/s} \) and \( \mu_r = 1 \):
\[ v = \frac{3 \times 10^8}{\sqrt{\epsilon_r}} \]

### Step 4: Set the two expressions for velocity equal
Now we can set the two expressions for \( v \) equal to each other:
\[ 10^8 = \frac{3 \times 10^8}{\sqrt{\epsilon_r}} \]
Cross-multiplying gives:
\[ 10^8 \sqrt{\epsilon_r} = 3 \times 10^8 \]
Dividing both sides by \( 10^8 \):
\[ \sqrt{\epsilon_r} = 3 \]

### Step 5: Solve for \( \epsilon_r \)
Squaring both sides:
\[ \epsilon_r = 9 \]

### Step 6: Relate \( \epsilon_r \) to \( x \)
We know that:
\[ \epsilon_r = \frac{1}{x} \]
Thus, we can set up the equation:
\[ \frac{1}{x} = 9 \]
Taking the reciprocal gives:
\[ x = \frac{1}{9} \]

### Final Answer:
\[ x = 9 \]