Question

A plane electromagnetic wave of frequency 35 MHz travels in free space along the X-direction. At a particular point (in space and time), \( \vec{E} = 9.6 \hat{j} \, \text{V/m} \). The value of the magnetic field at this point is:

• A. \( 3.2 \times 10^{-8} \hat{k} \, \text{T} \)
• B. \( 3.2 \times 10^{-8} \hat{i} \, \text{T} \)
• C. \( 9.6 \hat{j} \, \text{T} \)
• D. \( 9.6 \times 10^{-8} \hat{k} \, \text{T} \)

Answer 1

The Correct Option is A

The magnetic field accompanying an electric field in a plane electromagnetic wave is determined by the equation \(E = cB\), where \(E\) represents the electric field magnitude, \(B\) the magnetic field magnitude, and \(c\) the speed of light (approximately \(3 \times 10^8 \text{ m/s}\)).

Given the electric field vector as \( \vec{E} = 9.6 \hat{j} \, \text{V/m} \), and knowing the wave propagates in the X-direction, the magnetic field must be perpendicular to both the direction of propagation (X) and the electric field (Y). Consequently, the magnetic field lies along the Z-direction.

The magnitude of the magnetic field is calculated as:

\(B = \frac{E}{c} = \frac{9.6}{3 \times 10^8}\)

This yields:

\(B = 3.2 \times 10^{-8} \, \text{T}\)

Considering the Z-direction, the magnetic field vector is:

\(\vec{B} = 3.2 \times 10^{-8} \hat{k} \, \text{T}\)

This matches the correct option.

Alternative options are incorrect because:

• \(3.2 \times 10^{-8} \hat{i} \, \text{T}\): The magnetic field is not in the X-direction, as the wave propagates in this direction.
• \(9.6 \hat{j} \, \text{T}\): This option presents an incorrect unit and direction inconsistent with the derived relationship.
• \(9.6 \times 10^{-8} \hat{k} \, \text{T}\): The magnitude of the magnetic field is incorrect.

Therefore, the magnetic field at this point is \(3.2 \times 10^{-8} \hat{k} \, \text{T}\).