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

The given electric field for the wave is \( \vec{E} = 69 \sin(\omega t - kx)\,\hat{j} \). This indicates that the electric field oscillates in the \( \hat{j} \) (y-axis) direction.

According to the electromagnetic wave theory, in free space, the electric field (\( \vec{E} \)), magnetic field (\( \vec{B} \)), and the direction of wave propagation (\( \vec{k} \)) are mutually perpendicular. Therefore, the direction of propagation for this wave is along the \( x \)-axis as suggested by the term \( \omega t - kx \), where \( kx \) represents the direction of wave vector (\( k \hat{i} \)).

To determine the magnetic field direction, we use the right-hand rule related to electromagnetic waves: 

1. Point your thumb in the direction of wave propagation (\( \hat{i} \)).
2. Point your index finger in the direction of the electric field (\( \hat{j} \)).
3. Your middle finger, perpendicular to both, will point in the direction of the magnetic field.

Following this rule, if the thumb points along \( \hat{i} \) (direction of propagation) and the index finger points along \( \hat{j} \) (direction of the electric field), the middle finger will naturally point along the positive \( \hat{k} \) direction. Hence, the direction of the magnetic field is \( \hat{k} \).

So, the correct answer is the direction of the magnetic field is \(\hat{k}\).

Thus, the correct option is \( \hat{k} \).