Question

For a plane electromagnetic wave propagating in x-direction, which one of the following combination gives the correct possible directions for electric field (E) and magnetic field (B) respectively

• A. \(-\hat{j}+\hat{k},-\hat{j}+\hat{k}\)
• B. \(\hat{j}+\hat{k},\hat{j}+\hat{k}\)
• C. \(-\hat{j}+\hat{k},-\hat{j}-\hat{k}\)
• D. \(\hat{j}+\hat{k},-\hat{j}-\hat{k}\)

Answer 1

The Correct Option is C

To solve this problem, let us first understand the nature of electromagnetic (EM) waves:

• An electromagnetic wave propagates in space with both an electric field (\( \vec{E} \)) and a magnetic field (\( \vec{B} \)) perpendicular to each other and to the direction of propagation.
• For a plane electromagnetic wave traveling along the x-direction, the electric field (\( \vec{E} \)) and the magnetic field (\( \vec{B} \)) must be perpendicular to the x-axis and, consequently, to each other.
• The possible directions for \( \vec{E} \) and \( \vec{B} \) are in the y and z directions, represented by unit vectors \( \hat{j} \) and \( \hat{k} \).
• Using the right-hand rule, if we point our thumb in the direction of wave propagation (x-direction), our fingers should be able to curl from \( \vec{E} \) to \( \vec{B} \).

Now, let's evaluate the given options:

1. \(-\hat{j}+\hat{k}\), \(-\hat{j}+\hat{k}\):
   • Both vectors are not perpendicular to each other; hence, this option is incorrect.
2. \(\hat{j}+\hat{k}\), \(\hat{j}+\hat{k}\):
   • Similar to the first option, both vectors are the same and do not satisfy the conditions of perpendicularity and the right-hand rule.
3. \(-\hat{j}+\hat{k}\), \(-\hat{j}-\hat{k}\):
   • These vectors \( (-\hat{j}+\hat{k}) \) and \( (-\hat{j}-\hat{k}) \) are perpendicular to each other.
   • Using \( \vec{k} = \vec{i} \times \vec{E} \) rule (i.e., cross product), this option satisfies both the perpendicularity condition and the right-hand rule.
4. \(\hat{j}+\hat{k}\), \(-\hat{j}-\hat{k}\):
   • Although these vectors are perpendicular, they do not satisfy the right-hand rule.

Based on the reasoning, the correct answer is the third option: \(-\hat{j}+\hat{k}\), \(-\hat{j}-\hat{k}\). These directions ensure that \( \vec{E} \) and \( \vec{B} \) are perpendicular to each other and adhere to the propagation direction of the EM wave.