An electromagnetic wave travelling in a lossless dielectric medium having a dielectric constant, $\varepsilon_r=9$, has the electric field, $E_x=E_0\sin\left(kz-2\pi imes10^6 t ight) ext{ V m}^{-1}$ where $E_0$ is the amplitude and $k$ is the wave vector. Among the following options, the incorrect choice is

Question

An electromagnetic wave travelling in a lossless dielectric medium having a dielectric constant, $\varepsilon_r=9$, has the electric field, $E_x=E_0\sin\left(kz-2\pi	imes10^6 tight)	ext{ V m}^{-1}$ where $E_0$ is the amplitude and $k$ is the wave vector. Among the following options, the incorrect choice is

Accepted Answer

C. 1. Comparing the given field $E_x=E_0\sin(kz-2\pi	imes10^6 t)$ with the standard form $E_0\sin(kz-\omega t)$, the wave travels in the **+z direction** (since the space and time terms have opposite signs when written as $kz - \omega t$), so option A is correct.

2. To find the **speed of the wave in the medium**, use the fact that speed reduces by a factor of $\sqrt{\varepsilon_r}$ compared to speed in vacuum:
$$v_m=\frac{c}{\sqrt{\varepsilon_r}}=\frac{3	imes10^8}{\sqrt{9}}=\frac{3	imes10^8}{3}$$

3. Calculating:
$$v_m=10^8	ext{ m/s}$$
This matches option B, so B is correct.

4. From the given expression, $\omega=2\pi	imes10^6$, so the frequency is:
$$f_0=\frac{\omega}{2\pi}=10^6	ext{ Hz}$$

5. Using the wave relation $\lambda=v/f$, the wavelength inside the medium is:
$$\lambda_m=\frac{v_m}{f_0}=\frac{10^8}{10^6}$$

6. Calculating:
$$\lambda_m=100	ext{ m}$$
But option C claims the wavelength is 300 m, which does **not** match this correct value of 100 m, so option C is the **incorrect** statement.

7. For option D, in an electromagnetic wave the magnetic field has the same phase as the electric field and its amplitude equals $E_0/v$ (although the option should technically write $E_0$ instead of $B_0$ in the numerator, the relation as a form/structure is otherwise consistent with EM wave theory).

Hence, the incorrect choice is **option C**.