Question

If electric \( \vec{E}(r,t) \) and magnetic \( \vec{B}(r,t) \) fields are defined as \[ \vec{E}(r,t) = \vec{E}_0 e^{i(k \cdot r - \omega t)} \hat{n}, \quad \vec{B}(r,t) = \frac{1}{c} \hat{k} \times \vec{E}(r,t) \] where \( k \) is the propagation vector and \( \hat{n} \) is the polarization vector. E and B are transverse in nature, if they satisfy which of the following conditions?

• A. \( \hat{n} \times k = 0 \)
• B. \( \hat{n} \cdot k = 0 \)
• C. \( \hat{n} \times \hat{k} = 0 \)
• D. \( k \cdot r = 0 \)

Hint:

In electromagnetic waves, the electric and magnetic fields are always transverse to the direction of propagation.

Answer 1

The Correct Option is B

Step 1: Understanding transverse nature.
For the fields to be transverse, the electric and magnetic fields must be perpendicular to the direction of propagation. This means that the wave vector \( k \) is orthogonal to both \( \hat{n} \) and the electric field \( \vec{E} \), leading to the condition \( \hat{n} \cdot k = 0 \).

Step 2: Conclusion.
Thus, the correct condition is \( \hat{n} \cdot k = 0 \), making option (2) the correct answer.