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 electromagnetic wave propagating through free space, the electric field vector (\( \vec{E} \)), magnetic field vector (\( \vec{B} \)), and the direction of wave propagation (\( \vec{v} \)) are mutually perpendicular.
Step 2: Key Formula or Approach:
Two main relationships are used here:
1. Magnitude Relationship: \( E_0 = c B_0 \), where \( c = 3 \times 10^8 \text{ m/s} \).
2. Direction Relationship: The direction of wave propagation is given by the cross product of the electric and magnetic field directions: \( \hat{v} = \hat{E} \times \hat{B} \).
Step 3: Detailed Explanation:
First, find the magnitude of \( \vec{E} \):
Given \( B_0 = 1.2 \times 10^{-8} \text{ T} \).
Using the magnitude formula: \[ E_0 = c \times B_0 \] \[ E_0 = (3 \times 10^8 \text{ m/s}) \times (1.2 \times 10^{-8} \text{ T}) \] \[ E_0 = 3 \times 1.2 = 3.6 \text{ V/m} \]
Second, find the direction of \( \vec{E} \):
Given direction of propagation \( \hat{v} = \hat{i} \) (X-direction).
Given direction of magnetic field \( \hat{B} = \hat{k} \) (Z-direction).
Using the cross product relationship: \[ \hat{i} = \hat{E} \times \hat{k} \] From the standard right-hand rule for unit vectors, we know that \( \hat{j} \times \hat{k} = \hat{i} \).
Therefore, the direction of the electric field must be \( \hat{j} \).
Combining magnitude and direction:
\[ \vec{E} = E_0 \hat{E} = 3.6\hat{j} \]
Step 4: Final Answer:
The value of E at that point is \( 3.6\hat{j} \text{ Vm}^1 \).