Electromagnetic waves travel in a medium which has relative permeability `1.3` and relative permittivity `2.14`. Then the speed of the electromagnetic wave in the medium will be

Options

A. `1.36xx10^(6)m//s`
B. `1.8xx10^(2)m//s`
C. `3.6xx10^(8)m//s`
D. `1.8xx10^(8)m//s`

Explanation

To find the speed of electromagnetic waves in a medium with given relative permeability and relative permittivity, we can follow these steps:

### Step 1: Understand the relationship between speed, refractive index, and the properties of the medium.
The speed of electromagnetic waves in a medium (v) is given by the equation:
\[ v = \frac{c}{n} \]
where:
- \( c \) is the speed of light in vacuum (approximately \( 3 \times 10^8 \, \text{m/s} \)),
- \( n \) is the refractive index of the medium.

### Step 2: Calculate the refractive index (n).
The refractive index \( n \) can be calculated using the formula:
\[ n = \sqrt{\mu_r \epsilon_r} \]
where:
- \( \mu_r \) is the relative permeability (given as \( 1.3 \)),
- \( \epsilon_r \) is the relative permittivity (given as \( 2.14 \)).

Now, substituting the values:
\[ n = \sqrt{1.3 \times 2.14} \]

### Step 3: Perform the multiplication.
Calculate \( 1.3 \times 2.14 \):
\[ 1.3 \times 2.14 = 2.782 \]

### Step 4: Take the square root.
Now, take the square root of \( 2.782 \):
\[ n = \sqrt{2.782} \approx 1.67 \]

### Step 5: Calculate the speed of electromagnetic waves (v).
Now, substitute \( n \) back into the speed formula:
\[ v = \frac{c}{n} = \frac{3 \times 10^8 \, \text{m/s}}{1.67} \]

### Step 6: Perform the division.
Calculating the division:
\[ v \approx \frac{3 \times 10^8}{1.67} \approx 1.796 \times 10^8 \, \text{m/s} \]

### Step 7: Round the answer.
Round the result to two significant figures:
\[ v \approx 1.8 \times 10^8 \, \text{m/s} \]

### Final Answer:
The speed of the electromagnetic wave in the medium is approximately \( 1.8 \times 10^8 \, \text{m/s} \).

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