Question

The electric field of a plane electromagnetic wave in a medium is given by

where c is the speed of light in free space. E field is polarized in the x −z plane. The speed of the wave is v in the medium. Then:

• A. \( \hat{n} = \hat{i} - \hat{k}; \quad v = c \)
• B. \( \hat{n} = \frac{\hat{i} - \hat{k}}{\sqrt{2}}; \quad v = \frac{c}{\sqrt{3}} \)
• C. Refractive index of the medium is \(\sqrt{3}\)
• D. \( \hat{n} = \frac{\hat{i} + \hat{k}}{\sqrt{2}}; \quad v = \frac{c}{2} \)

Hint:

In electromagnetic waves, the wave vector direction k determines the direction of prop agation, and the polarization direction ˆn is perpendicular to both k and the direction of the electric field.

Answer 1

The Correct Option is B, C

1. Step 1: The electric field of a plane electromagnetic wave is described by:

   \( E(x, y, z, t) = E_0 \hat{n} e^{i\mathbf{k} \cdot \mathbf{r} - i\omega t} \)

   Where:

   • \( \mathbf{k} \) is the wave vector.
   • \( \mathbf{r} = (x, y, z) \).
   • \( \omega \) is the angular frequency.
2. Step 2: The wave vector \( \mathbf{k} \) indicates the wave's propagation direction. The specific wave expression is:

   \( e^{ik_0(x + y + z) - i\omega t} \)

   Therefore, the wave vector is:

   \( \mathbf{k} = k_0 (\hat{i} + \hat{j} + \hat{k}) \)

   Here, \( k_0 \) is the magnitude of the wave vector.

3. Step 3: The electric field is polarized in the \( x - z \) plane. Consequently, the unit vector \( \hat{n} \) must be orthogonal to the wave vector \( \mathbf{k} \). This is mathematically represented by a cross product of zero.
4. Step 4: Solving for \( \hat{n} \), we obtain:

   \( \hat{n} = \frac{\hat{i} - \hat{k}}{\sqrt{2}} \)

   The wave's speed in the medium is given by:

   \( v = \frac{c}{\sqrt{\varepsilon_r \mu_r}} \)

   Where \( c \) is the speed of light, \( \varepsilon_r \) is the relative permittivity, and \( \mu_r \) is the relative permeability of the medium.

Conclusion: The electric field polarization is \( \hat{n} = \frac{\hat{i} - \hat{k}}{\sqrt{2}} \), and the wave propagates at a speed \( v \) dependent on the medium's permittivity and permeability.