Question

A linearly polarized electromagnetic wave in vacuum is \[ E = 3.1 \cos [(1.8)z - (5.4 \times 10^8)t] \hat{i} \text{ N/C} \] is incident normally on a perfectly reflecting wall at $z=a$. Choose the correct option :

• A. The frequency of electromagnetic wave is $54 \times 10^4$ Hz.
• B. The reflected wave will be $3.1 \cos [(1.8)z + (5.4 \times 10^8)t] \hat{i}$ N/C
• C. The transmitted wave will be $3.1 \cos [(1.8)z - (5.4 \times 10^8)t] \hat{i}$ N/C
• D. The wavelength is 5.4 m

Hint:

For a wave \(\cos(kx - \omega t)\), if the signs of the \(x\) and \(t\) terms are different, it moves in the positive direction. If they are the same, it moves in the negative direction.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
An electromagnetic wave incident on a perfectly reflecting wall is reflected back.
If the incident wave travels in the \(+z\) direction, the reflected wave must travel in the \(-z\) direction.
A wave travelling in \(-z\) direction has the form \(\cos(kz + \omega t)\).
Step 2: Key Formula or Approach:
General wave equation: \(E = E_0 \cos(kz - \omega t)\) for \(+z\) propagation.
Reflected wave equation: \(E_r = E_0 \cos(kz + \omega t + \phi)\).
Step 3: Detailed Explanation:
Given \(E = 3.1 \cos [(1.8)z - (5.4 \times 10^8)t] \hat{i}\).
Here, \(k = 1.8 \text{ m}^{-1}\) and \(\omega = 5.4 \times 10^8 \text{ rad/s}\).

Check Option A: \(f = \omega / 2\pi = (5.4 \times 10^8) / (2 \times 3.14) \approx 8.6 \times 10^7 \text{ Hz}\). Incorrect.
Check Option D: \(\lambda = 2\pi / k = 2\pi / 1.8 \approx 3.49 \text{ m}\). Incorrect.
Check Option C: Since the wall is perfectly reflecting, there is no transmitted wave. Incorrect.
Check Option B: The reflected wave propagates in the opposite direction. Changing the sign of the \(t\) term relative to \(z\) (or vice-versa) changes the direction of propagation. So, \(\cos(1.8z + 5.4 \times 10^8 t)\) correctly represents a wave moving in the \(-z\) direction.
Step 4: Final Answer:
The reflected wave is \(3.1 \cos [(1.8)z + (5.4 \times 10^8)t] \hat{i}\) N/C.