Question

Consider a rectangular coordinate system \( (x, y, z) \) with unit vectors \( \hat{a}_x, \hat{a}_y, \hat{a}_z \). A plane wave traveling in the region \( z \geq 0 \) with electric field vector \[ E = 10 \cos(2 \times 10^8 t + \beta z) \hat{a}_y \] \text{is incident normally on the plane at } z = 0, \text{ where } \beta \text{ is the phase constant. The region } z \geq 0 \text{ is in free space and the region } z<0 \text{ is filled with a lossless medium (permittivity } \varepsilon = \varepsilon_0, \text{ permeability } \mu = 4\mu_0, \text{ where } \varepsilon_0 = 8.85 \times 10^{-12} \text{ F/m and } \mu_0 = 4\pi \times 10^{-7} \text{ H/m). The value of the reflection coefficient is:}

• A. \( \frac{1}{3} \)
• B. \( \frac{3}{5} \)
• C. \( \frac{2}{5} \)
• D. \( \frac{2}{3} \)

Hint:

To calculate the reflection coefficient, use the intrinsic impedances of the media on either side of the boundary. The formula is \( R = \frac{Z_2 - Z_1}{Z_2 + Z_1} \), where \( Z_1 \) and \( Z_2 \) are the impedances of the two media.

Answer 1

The Correct Option is A

We are given a plane wave traveling in the region \( z \geq 0 \), and the electric field vector is provided. The problem involves calculating the reflection coefficient at the boundary \( z = 0 \), where the medium changes. Step 1: Identify the reflection coefficient formula.
The reflection coefficient \( R \) is given by the equation: \[ R = \frac{Z_2 - Z_1}{Z_2 + Z_1}, \] where \( Z_1 \) and \( Z_2 \) are the intrinsic impedances of the media in the regions \( z<0 \) and \( z \geq 0 \), respectively. Step 2: Calculate the impedances of the media.
The impedance of free space \( Z_1 \) is given by: \[ Z_1 = \sqrt{\frac{\mu_0}{\varepsilon_0}}. \] For the second medium (with \( \mu = 4\mu_0 \) and \( \varepsilon = \varepsilon_0 \)), the impedance \( Z_2 \) is: \[ Z_2 = \sqrt{\frac{\mu}{\varepsilon}} = \sqrt{\frac{4\mu_0}{\varepsilon_0}} = 2Z_1. \] Step 3: Calculate the reflection coefficient.
Substituting the values into the reflection coefficient formula, we get: \[ R = \frac{2Z_1 - Z_1}{2Z_1 + Z_1} = \frac{Z_1}{3Z_1} = \frac{1}{3}. \] Final Answer: \[ \boxed{\frac{1}{3}}. \]