Question

A plane electromagnetic wave travels in free space along X-direction. If the value of B (in tesla) at a particular point in space and time is \( 1.2 \times 10^{-8} \hat{k} \). The value of E (in Vm⁻¹) at that point is

• A. \( 1.2 \hat{j} \)
• B. \( 3.6 \hat{k} \)
• C. \( 1.2 \hat{k} \)
• D. \( 3.6 \hat{j} \)
• E. \( 0.4 \hat{i} \)

Hint:

Use the cyclic mnemonic for cross products: \( i \to j \to k \to i \). Since the wave goes to \( i \) and \( B \) is in \( k \), \( E \) must be in \( j \) because \( j \times k = i \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In an EM wave, the electric field (\( \vec{E} \)), magnetic field (\( \vec{B} \)), and the direction of propagation are mutually perpendicular. Their magnitudes are related by the speed of light (\( c \)).
Step 2: Key Formula or Approach:
1. Magnitude relation: \( E = B \times c \).
2. Direction relation: The direction of \( \vec{E} \times \vec{B} \) gives the direction of propagation.
Step 3: Detailed Explanation:
1. Calculate the magnitude of \( E \): \[ E = (1.2 \times 10^{-8} \text{ T}) \times (3 \times 10^8 \text{ m/s}) \] \[ E = 1.2 \times 3 = 3.6 \text{ V/m} \] 2. Determine the direction: - Propagation is along \( +\hat{i} \). - \( \vec{B} \) is along \( +\hat{k} \). - We need \( \hat{Direction_E} \times \hat{k} = \hat{i} \). - Following the cross-product rule (\( \hat{j} \times \hat{k} = \hat{i} \)), the direction of \( \vec{E} \) must be along \( \hat{j} \). 3. Combining magnitude and direction: \( \vec{E} = 3.6 \hat{j} \).
Step 4: Final Answer:
The value of E at that point is \( 3.6 \hat{j} \).