Question

At any instant, if \( \vec{B} = -2 \times 10^{-7} \hat{j} \, \text{T} \) and \( \vec{C} \) is along the +x axis, then \( \vec{E} \) at this instant is:

• A. \(60 \hat{k}  \text{V/m} \)
• B. \(45 \hat{k}  \text{V/m} \)
• C. 90 \( \hat{k}  \text{V/m} \)
• D. 30 \( \hat{k}  \text{V/m} \)

Hint:

Remember the right-hand rule when calculating the cross-product of vectors, especially in electromagnetic wave propagation, where the direction of \( \vec{E} \), \( \vec{B} \), and the propagation direction are always mutually perpendicular.

Answer 1

The Correct Option is A

Step 1: Understanding the situation.
The magnetic field is given as \( \vec{B} = -2 \times 10^{-7} \hat{j} \, \text{T} \). We need to determine the electric field \( \vec{E} \) using the relationship between the electric and magnetic fields in an electromagnetic wave. The wave is propagating along the +x direction, which will be used to determine the direction of the electric field using the right-hand rule.
Step 2: Relation between electric and magnetic fields.
In an electromagnetic wave, the electric field is related to the magnetic field and the speed of light by the following vector relation: \[ \vec{E} = c \, \hat{i} \times \vec{B} \] where \( c \) is the speed of light \( (c = 3 \times 10^8 \, \text{m/s}) \), and \( \hat{i} \) represents the unit vector in the direction of wave propagation (along the +x axis).
Step 3: Calculation of the electric field.
Substituting the given magnetic field: \[ \vec{E} = c \, \hat{i} \times (-2 \times 10^{-7} \hat{j}) \] Using the cross-product rule \( \hat{i} \times \hat{j} = \hat{k} \): \[ \vec{E} = 3 \times 10^8 \times (-2 \times 10^{-7}) \hat{k} \] \[ \vec{E} = 60 \, \hat{k} \, \text{V/m} \] Thus, the electric field is directed along the positive z-axis.
Final Answer: \( 60 \, \hat{k} \, \text{V/m} \).