Question

A wave propagates whose electric field is given by \[ \vec{E} = 69 \sin(\omega t - kx)\,\hat{j}. \] Find the direction of the magnetic field.

• A. \( \hat{k} \)
• B. \( -\hat{k} \)
• C. \( \dfrac{\hat{i}+\hat{j}}{\sqrt{2}} \)
• D. \( \dfrac{\hat{i}-\hat{j}}{\sqrt{2}} \)

Hint:

For electromagnetic waves:
Direction of propagation \( \vec{k} = \vec{E} \times \vec{B} \)
\( \vec{E}, \vec{B}, \vec{k} \) form a right-handed orthogonal set
Phase term \( (\omega t - kx) \) indicates propagation along \(+x\)

Answer 1

The Correct Option is A

Step 1: The given electric field is: \[ \vec{E} = 69 \sin(\omega t - kx)\,\hat{j} \] which shows that the electric field oscillates along the \(+\hat{j}\) (y-axis).
Step 2: The phase \((\omega t - kx)\) indicates that the wave is propagating in the \(+\hat{i}\) (x-axis) direction.
Step 3: For an electromagnetic wave: \[ \vec{E} \perp \vec{B} \perp \vec{k} \] and the direction of propagation is given by: \[ \vec{k} = \vec{E} \times \vec{B} \]
Step 4: Since \[ \vec{E} \parallel \hat{j}, \vec{k} \parallel \hat{i}, \] we must have: \[ \hat{j} \times \vec{B} = \hat{i} \] This is satisfied when: \[ \vec{B} \parallel \hat{k} \]