Question

The electric field part of an electromagnetic wave in a medium is represented by: $E _{ x }=0$ $E _{ y }=2.5 \frac{ N }{ C } \cos \left[\left(2 \pi \times 10^{6} \frac{ rad }{ m }\right) t -\left(\pi \times 10^{-2} \frac{ rad }{ s }\right) x \right]$ $E _{ z }=0 .$ The wave is :

• A. moving along $x$ direction with frequency $10^6\, Hz$ and wavelength $100\, m$
• B. moving along $x$ direction with frequency $10^6\, Hz$ and wavelength $200\, m$
• C. moving along $- x$ direction with frequency $10^6\, Hz$ and wavelength $200\, m$
• D. moving along $y$ direction with frequency $2\pi \times 10^{6}\,Hz$ and wavelength $200\, m$

Answer 1

The Correct Option is B

To determine the characteristics of the wave, we examine the given electric field expression:

E_y = 2.5 \frac{N}{C} \cos \left[\left(2 \pi \times 10^{6} \frac{rad}{m}\right) t -\left(\pi \times 10^{-2} \frac{rad}{s}\right) x \right]

This is in the standard form of the wave equation:

E(y, t) = E_0 \cos(\omega t + kx + \phi)

where:

• \omega is the angular frequency
• k is the wave number
• \phi is the phase constant
• The wave propagates in the x direction.

From the given equation, we identify:

• \omega = 2\pi \times 10^{6} \, rad/s
• k = \pi \times 10^{-2} \, rad/m

The frequency f is derived from the angular frequency:

f = \frac{\omega}{2\pi} = \frac{2\pi \times 10^{6}}{2\pi} = 10^{6} \, Hz

The wavelength \lambda can be determined from the wave number:

\lambda = \frac{2\pi}{k} = \frac{2\pi}{\pi \times 10^{-2}} = 200 \, m

Since the cosine term is of the form \( \cos(\omega t - kx) \), it implies the wave is propagating in the positive x direction.

Therefore, the wave is moving along the x direction with frequency 10^{6} \, Hz and wavelength 200 \, m.

The correct answer is: moving along x direction with frequency 10^6 \, Hz and wavelength 200 \, m.