Question

Electromagnetic waves travel in a medium with a speed of \( 1.5 \times 10^8 \, \text{ms}^{-1} \). The relative permeability of the medium is \( 2.0 \). The relative permittivity will be:

• A. 5
• B. 4
• C. 1
• D. 2

Answer 1

The Correct Option is D

The speed of electromagnetic waves in a medium is related to its relative permeability (\( \mu_r \)) and relative permittivity (\( \varepsilon_r \)) by the equation:

\[ \varepsilon_r \mu_r = \frac{c^2}{v^2}, \]

where:
- \( c = 3 \times 10^8 \, \text{ms}^{-1} \) (speed of light in vacuum),
- \( v = 1.5 \times 10^8 \, \text{ms}^{-1} \) (speed of light in the medium),
- \( \mu_r = 2.0 \) (relative permeability of the medium).

Substituting the given values:

\[ \varepsilon_r \times 2 = \frac{(3 \times 10^8)^2}{(1.5 \times 10^8)^2}. \]

Simplify:

\[ \varepsilon_r \times 2 = \frac{9 \times 10^{16}}{2.25 \times 10^{16}}. \]

\[ \varepsilon_r \times 2 = 4. \]

\[ \varepsilon_r = 2. \]

Final Answer: \( \varepsilon_r = 2 \) (Option 4)

Answer 2

To find the relative permittivity of the medium, we begin with the relationship between the speed of electromagnetic waves, permittivity, and permeability. The speed of electromagnetic waves in a medium is given by:

\(v = \frac{c}{\sqrt{\mu_r \cdot \epsilon_r}}\)

where:

• \(v\) is the speed of electromagnetic waves in the medium (here, \(1.5 \times 10^8 \, \text{ms}^{-1}\)).
• \(c\) is the speed of light in vacuum (approximately \(3 \times 10^8 \, \text{ms}^{-1}\)).
• \(\mu_r\) is the relative permeability of the medium (given as \(2.0\)).
• \(\epsilon_r\) is the relative permittivity of the medium (unknown).

We need to find \(\epsilon_r\). Rearrange the formula to solve for \(\epsilon_r\):

\(\epsilon_r = \frac{c^2}{v^2 \cdot \mu_r}\)

Substitute the known values:

\(\epsilon_r = \frac{(3 \times 10^8)^2}{(1.5 \times 10^8)^2 \cdot 2}\)

Calculate the squared terms:

• \(c^2 = (3 \times 10^8)^2 = 9 \times 10^{16}\)
• \(v^2 = (1.5 \times 10^8)^2 = 2.25 \times 10^{16}\)

Substitute back into the formula:

\(\epsilon_r = \frac{9 \times 10^{16}}{2.25 \times 10^{16} \cdot 2}\)

Simplify the expression:

\(\epsilon_r = \frac{9}{4.5} = 2\)

Thus, the relative permittivity of the medium is \(2\).

The correct answer is: 2