Question

An electromagnetic wave travels in free space along the x-direction. At a particular point in space and time, \( B = 2 \times 10^{-7} \hat{j} \) T is associated with this wave. The value of the corresponding electric field \( E \) at this point is _______ V/m.}

• A. \( 60 \hat{k} \)
• B. \( 60 \hat{k} \)
• C. \( 30 \hat{k} \)
• D. \( 30 \hat{i} \)

Answer 1

The Correct Option is C

Step 1: Using the relation between electric and magnetic fields.
In an electromagnetic wave, the electric field \( \vec{E} \) and the magnetic field \( \vec{B} \) are related by the following equation: \[ \vec{E} = c \vec{B} \times \hat{n} \] where \( c = 3 \times 10^8 \, \text{m/s} \) is the speed of light, and \( \hat{n} \) is the unit vector in the direction of wave propagation (along the \( x \)-direction in this case).
Step 2: Calculating the electric field.
Given \( \vec{B} = 2 \times 10^{-7} \hat{j} \) and the wave travels in the \( x \)-direction, the cross product \( \vec{B} \times \hat{i} \) gives the direction of \( \vec{E} \). Therefore, we have: \[ \vec{E} = c \times 2 \times 10^{-7} \hat{k} = (3 \times 10^8) \times (2 \times 10^{-7}) \hat{k} = 60 \hat{k} \, \text{V/m} \]
Step 3: Conclusion.
The corresponding electric field \( \vec{E} \) is \( 60 \hat{k} \) V/m.
Final Answer: (C) \( 30 \hat{k} \)