Question

The expression for the magnetic field that induces the electric field \( \vec{E} = K(y\hat{i} + 3z\hat{j} + 4y\hat{k}) \cos(\omega t) \) is:

• A. \( -\frac{K}{\omega} (\hat{i} + y\hat{j} - z\hat{k}) \sin(\omega t) \)
• B. \( -\frac{K}{\omega} (\hat{i} + y\hat{j} + z\hat{k}) \sin(\omega t) \)
• C. \( \frac{K}{\omega} (\hat{i} - y\hat{j} + z\hat{k}) \sin(\omega t) \)
• D. \( \frac{K}{\omega} (\hat{i} + y\hat{j} + z\hat{k}) \sin(\omega t) \)

Hint:

Use \( \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \) to find the induced magnetic field from a time-varying electric field.

Answer 1

The Correct Option is A

Step 1: Use Maxwell’s equation.
From Faraday’s law, \[ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}. \] Step 2: Compute curl of \( \vec{E} \).
\[ \vec{E} = K(y\hat{i} + 3z\hat{j} + 4y\hat{k}) \cos(\omega t). \] \[ \nabla \times \vec{E} = K\cos(\omega t)[(\partial_y 4y - \partial_z 3z)\hat{i} + (\partial_z y - \partial_x 4y)\hat{j} + (\partial_x 3z - \partial_y y)\hat{k}] = K\cos(\omega t)(\hat{i} + \hat{j} + \hat{k}). \] Step 3: Relate to \( \vec{B} \).
\[ -\frac{\partial \vec{B}}{\partial t} = K(\hat{i} + \hat{j} + \hat{k}) \cos(\omega t). \] Integrate w.r.t. time: \[ \vec{B} = -\frac{K}{\omega}(\hat{i} + \hat{j} + \hat{k}) \sin(\omega t). \] Step 4: Final Answer.
Hence, \( \vec{B} = -\frac{K}{\omega} (\hat{i} + \hat{j} + \hat{k}) \sin(\omega t). \)