Question

A plane electromagnetic wave of wavelength $\lambda$ has an intensity I. It is propagating along the positive Y-direction. The allowed expressions for the electric and magnetic fields are given by :

• A. $\vec{E} = \sqrt{\frac{2I}{\epsilon_{o}c}} \cos \left[\frac{2\pi}{\lambda} \left(y -ct\right)\right]\hat{k} ; \vec{B} = + \frac{1}{c} E \hat{i} $
• B. $\vec{E} = \sqrt{\frac{2I}{\epsilon_{o}c}} \cos \left[\frac{2\pi}{\lambda} \left(y + ct\right)\right]\hat{k} ; \vec{B} = \frac{1}{c} E \hat{i} $
• C. $\vec{E} = \sqrt{\frac{I}{\epsilon_{o}c}} \cos \left[\frac{2\pi}{\lambda} \left(y - ct\right)\right]\hat{k} ; \vec{B} = \frac{1}{c} E \hat{i} $
• D. $\vec{E} = \sqrt{\frac{I}{\epsilon_{o}c}} \cos \left[\frac{2\pi}{\lambda} \left(y - ct\right)\right]\hat{i} ; \vec{B} = \frac{1}{c} E \hat{k} $

Answer 1

The Correct Option is A

To solve this problem, we need to determine the correct expressions for the electric field \(\vec{E}\) and the magnetic field \(\vec{B}\) for a plane electromagnetic wave propagating along the positive Y-direction. We are given four options to consider, each with different expressions for these fields.

Step 1: Understanding the Direction of Propagation

An electromagnetic wave propagating along the positive Y-direction has its electric field (\(\vec{E}\)) and magnetic field (\(\vec{B}\)) perpendicular to the direction of propagation and to each other. This forms the basis of electromagnetic wave propagation.

Step 2: Amplitude of the Electric Field

The intensity (\(I\)) of a plane electromagnetic wave is related to the amplitude of the electric field by the formula:

\(I = \frac{1}{2} \epsilon_0 c E_0^2\),

where \(E_0\) is the amplitude of the electric field, \(\epsilon_0\) is the permittivity of free space, and \(c\) is the speed of light. Solving for \(E_0\), we get:

\(E_0 = \sqrt{\frac{2I}{\epsilon_0 c}}\)

Step 3: Selecting the Correct Option

Given that the wave is propagating in the positive Y-direction, the wave vector \(k\) should be in the form \(\cos\left[\frac{2\pi}{\lambda} \left(y - ct\right)\right]\), representing a wave traveling in the positive Y-direction.

Considering the given options, the correct one should yield \(E_0\) as follows:

1. The electric field \(\vec{E}\) must be perpendicular to Y, hence \(\hat{k}\) or \(\hat{i}\) are valid directions.
2. The magnetic field \(\vec{B}\) must be perpendicular to both the direction of propagation and \(\vec{E}\).

Upon evaluating the options, the correct answer that satisfies these conditions is:

\(\vec{E} = \sqrt{\frac{2I}{\epsilon_{o}c}} \cos \left[\frac{2\pi}{\lambda} \left(y -ct\right)\right]\hat{k} ; \vec{B} = + \frac{1}{c} E \hat{i}\)

Conclusion: This option correctly represents the orientation and propagation for the electric and magnetic fields of a plane wave traveling along the positive Y-direction, with the electric field perpendicular to it along the Z-axis (\( \hat{k} \)), and the magnetic field along the X-axis (\( \hat{i} \)).