An electromagnetic wave is travelling along z-axis and its electric f…

The correct answer is Option (4) → \( \hat{k}\times\vec{E} = 0,\ \hat{k}\times\vec{B} = 0 \)

An electromagnetic wave is transverse: \( \vec{E} \), \( \vec{B} \) and the direction of travel \( \hat{k} \) are mutually perpendicular, with \( \vec{E}\times\vec{B} \) along \( \hat{k} \) and \( |\vec{E}| = c|\vec{B}| \). The phase \( \omega t - kz \) shows that the wave travels along +z.

Option (1) \( \vec{B} = \displaystyle\frac{1}{c}\hat{k}\times\vec{E} \) — Correct. \( \hat{k}\times\vec{E} \) is perpendicular to both the direction of travel and \( \vec{E} \), with magnitude E; dividing by c gives exactly \( \vec{B} \).

Option (2) \( \vec{E} = c\,(\vec{B}\times\hat{k}) \) — Correct. Putting \( \vec{B} = \frac{1}{c}\hat{k}\times\vec{E} \): \( (\hat{k}\times\vec{E})\times\hat{k} = \vec{E}(\hat{k}\cdot\hat{k}) - \hat{k}(\vec{E}\cdot\hat{k}) = \vec{E} \), because \( \vec{E} \) is perpendicular to \( \hat{k} \). So \( c(\vec{B}\times\hat{k}) = \vec{E} \).

Option (3) \( \hat{k}\cdot\vec{E} = 0,\ \hat{k}\cdot\vec{B} = 0 \) — Correct. A zero dot product means the fields are perpendicular to the direction of propagation, which is what transverse means.

Option (4) \( \hat{k}\times\vec{E} = 0,\ \hat{k}\times\vec{B} = 0 \) — Incorrect. A cross product is zero only for parallel vectors. \( \vec{E} \) and \( \vec{B} \) are perpendicular to \( \hat{k} \), so \( |\hat{k}\times\vec{E}| = E \) and \( |\hat{k}\times\vec{B}| = B \), not zero. This option would describe a longitudinal wave.

NCERT (2026-27):
Nature of Electromagnetic Waves – “It can be shown from Maxwell’s equations that electric and magnetic fields in an electromagnetic wave are perpendicular to each other, and to the direction of propagation.”