Magnetic Field Vector Representation in Electromagnetic Wave

For a plane electromagnetic wave propagating in vacuum, the electric field $\vec{E}$ and magnetic field $\vec{B}$ can be represented in complex exponential form as: $\vec{E}(\vec{r}, t) = \vec{E}_0 e^{i(\vec{k} \cdot \vec{r} - \omega t)}$ $\vec{B}(\vec{r}, t) = \vec{B}_0 e^{i(\vec{k} \cdot \vec{r} - \omega t)}$

According to Maxwell's third equation (Faraday's Law of Induction) in a charge-free vacuum: $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$

Applying the differential operators to the plane wave functions:

1. The curl of the electric field yields: $\nabla \times \vec{E} = i \vec{k} \times \vec{E}$

2. The negative partial time derivative of the magnetic field yields: $-\frac{\partial \vec{B}}{\partial t} = -(-i\omega \vec{B}) = i\omega \vec{B}$

Equating both sides from Faraday's Law: $i \vec{k} \times \vec{E} = i \omega \vec{B}$

Dividing both sides by $i\omega$: $\vec{B} = \frac{\vec{k} \times \vec{E}}{\omega}$

Thus, the magnetic field associated with the electromagnetic wave is given by: $\frac{\vec{k} \times \vec{E}}{\omega}$

Correct Option: B