Question

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

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

Hint:

In electromagnetic waves, the electric and magnetic fields are always perpendicular to each other and to the direction of wave propagation.

Answer 1

The Correct Option is A

To determine the direction of the magnetic field (\( \mathbf{B} \)) in a wave where the electric field (\( \mathbf{E} \)) is given by \(\mathbf{E} = 69 \sin(\omega t - kx) \hat{j}\), we need to apply the principles of electromagnetic (EM) wave propagation. In an electromagnetic wave, the electric field (\( \mathbf{E} \)), magnetic field (\( \mathbf{B} \)), and the direction of wave propagation (\( \mathbf{k} \)) are all perpendicular to each other. This relationship can be expressed using the right-hand rule of cross products.

For electromagnetic waves:

1. The direction of wave propagation is along the positive x-axis (denoted by \(\hat{i}\)).
2. The electric field vector \(\mathbf{E}\) is given as being in the \(\hat{j}\) direction.
3. According to the right-hand rule: \( \mathbf{k} = \mathbf{E} \times \mathbf{B} \). So, if you point your right thumb in the direction of \(\mathbf{E} \) (\(\hat{j}\)) and the wave propagation along \( x \) (\(\hat{i}\)), your fingers will curl in the direction of \(\mathbf{B}\).

Given that \( \mathbf{E} \) is along \(\hat{j}\) and the wave is propagating along the \(\hat{i}\) direction, the cross product \(\mathbf{E} \times \mathbf{B}\) should follow the propagation direction. For the cross product to result in \(\hat{i}\), the magnetic field should be directed along the \(\hat{k}\), as:

\(\mathbf{E}(\hat{j})\)

\(\times \mathbf{B}(\hat{k})\)

= \(\hat{i}\)

Therefore, the direction of the magnetic field \(\mathbf{B}\) is \(\hat{k}\).

Let's verify the correct option: \(\hat{k}\), since it satisfies the conditions described above.