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.
We are given the magnetic field \( \vec{B} = -2 \times 10^{-7} \hat{j} \, \text{T} \) and the electric field \( \vec{E} \) is to be calculated using the relation between the electric and magnetic fields in an electromagnetic wave. The direction of \( \vec{C} \) is along the +x axis, which is relevant for applying the right-hand rule.
Step 2: Applying the cross-product for \( \vec{E} \).
The electric field in the electromagnetic wave is related to the magnetic field and the velocity of light using the following equation: \[ \vec{E} = c \, \hat{k} \times \vec{B} \] where \( c \) is the speed of light (\( c = 3 \times 10^8 \, \text{m/s} \)) and \( \hat{k} \) is the unit vector in the direction of the wave's propagation (which is along the +x axis).
Step 3: Calculation.
We calculate the cross-product \( \hat{k} \times \hat{j} \), which results in \( \hat{i} \) (direction of propagation along the x-axis): \[ \vec{E} = c \, \hat{k} \times (-2 \times 10^{-7} \hat{j}) = 60 \, \hat{k} \, \text{V/m} \] This matches the correct answer option. Final Answer: 60 \( \hat{k} \, \text{V/m} \).