Question

An electromagnetic wave travelling in a lossless dielectric medium having a dielectric constant, \[ \varepsilon_r = 9, \] has the electric field \[ E_x=E_0\sin(kz-2\pi\times10^6 t)\ \text{V m}^{-1} \] where \(E_0\) is the amplitude and \(k\) is the wave vector. Among the following options, the incorrect choice is:

• A. The direction of propagation of the electromagnetic wave is along \(+z\)
• B. The speed of the electromagnetic wave inside the medium is \(10^8\,\text{m s}^{-1}\)
• C. The wavelength of the electromagnetic wave inside the medium is \(300\,\text{m}\)
• D. The magnetic field is given by \[ B_y=\frac{E_0}{v}\sin(kz-2\pi\times10^6t) \]

Hint:

Remember: \[ v=\frac{c}{\sqrt{\varepsilon_r}} \] and \[ \lambda=\frac{v}{f} \] In a dielectric medium, wavelength changes but frequency remains unchanged.

Answer 1

The Correct Option is C

Concept:

• Electromagnetic wave: \[ E=E_0\sin(kz-\omega t) \] propagates in the \(+z\)-direction.

• Speed in a dielectric medium: \[ v=\frac{c}{\sqrt{\varepsilon_r}} \]

• Frequency: \[ f=\frac{\omega}{2\pi} \]

• Wavelength: \[ \lambda=\frac{v}{f} \]

Step 1: Determine direction of propagation
Given, \[ E_x=E_0\sin(kz-\omega t) \] Since the phase is \(kz-\omega t\), \[ \boxed{\text{Wave propagates along }+z} \] Hence option (A) is correct.

Step 2: Calculate wave speed
Given, \[ \varepsilon_r=9 \] Therefore, \[ v=\frac{c}{\sqrt{\varepsilon_r}} \] \[ v=\frac{3\times10^8}{3} \] \[ v=10^8\,\text{m s}^{-1} \] Hence option (B) is correct.

Step 3: Calculate frequency
Comparing, \[ \omega=2\pi\times10^6 \] Thus, \[ f=\frac{\omega}{2\pi} \] \[ f=10^6\,\text{Hz} \]

Step 4: Calculate wavelength
\[ \lambda=\frac{v}{f} \] \[ \lambda=\frac{10^8}{10^6} \] \[ \lambda=100\,\text{m} \] Therefore wavelength is not \(300\,\text{m}\). Hence option (C) is incorrect.

Step 5: Check magnetic field relation
For an EM wave, \[ E=vB \] Thus, \[ B=\frac{E}{v} \] \[ B_y=\frac{E_0}{v}\sin(kz-\omega t) \] Hence option (D) is correct. \[ \boxed{\text{Incorrect option = (C)}} \]