Question:medium
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:
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:
Show Hint
Remember:
\[
v=\frac{c}{\sqrt{\varepsilon_r}}
\]
and
\[
\lambda=\frac{v}{f}
\]
In a dielectric medium, wavelength changes but frequency remains unchanged.
Updated On: Jun 21, 2026
- The direction of propagation of the electromagnetic wave is along \(+z\)
- The speed of the electromagnetic wave inside the medium is \(10^8\,\text{m s}^{-1}\)
- The wavelength of the electromagnetic wave inside the medium is \(300\,\text{m}\)
- The magnetic field is given by \[ B_y=\frac{E_0}{v}\sin(kz-2\pi\times10^6t) \]
Show Solution
The Correct Option is C
Solution and Explanation
Step 1: Read the wave.
The field is $E_x = E_0\sin(kz - 2\pi\times10^6 t)$ in a medium with $\varepsilon_r = 9$. We must find the incorrect statement.
Step 2: Direction of travel.
A phase of the form $(kz - \omega t)$ means the wave moves along $+z$, so statement (A) is correct.
Step 3: Speed in the medium.
\[ v = \frac{c}{\sqrt{\varepsilon_r}} = \frac{3\times10^8}{\sqrt{9}} = \frac{3\times10^8}{3} = 10^8\,\text{m s}^{-1} \]
So statement (B) is correct.
Step 4: Frequency.
From $\omega = 2\pi\times10^6$, the frequency is $f = \omega/2\pi = 10^6\,\text{Hz}$.
Step 5: Wavelength.
\[ \lambda = \frac{v}{f} = \frac{10^8}{10^6} = 100\,\text{m} \]
The claimed value $300\,\text{m}$ is wrong, so statement (C) is the incorrect one.
Step 6: Magnetic field check.
For an EM wave $E = vB$, giving $B_y = \dfrac{E_0}{v}\sin(kz - \omega t)$, so statement (D) is correct. The incorrect choice is (C).
\[ \boxed{\text{The wavelength inside the medium is } 300\,\text{m (incorrect)}} \]
The field is $E_x = E_0\sin(kz - 2\pi\times10^6 t)$ in a medium with $\varepsilon_r = 9$. We must find the incorrect statement.
Step 2: Direction of travel.
A phase of the form $(kz - \omega t)$ means the wave moves along $+z$, so statement (A) is correct.
Step 3: Speed in the medium.
\[ v = \frac{c}{\sqrt{\varepsilon_r}} = \frac{3\times10^8}{\sqrt{9}} = \frac{3\times10^8}{3} = 10^8\,\text{m s}^{-1} \]
So statement (B) is correct.
Step 4: Frequency.
From $\omega = 2\pi\times10^6$, the frequency is $f = \omega/2\pi = 10^6\,\text{Hz}$.
Step 5: Wavelength.
\[ \lambda = \frac{v}{f} = \frac{10^8}{10^6} = 100\,\text{m} \]
The claimed value $300\,\text{m}$ is wrong, so statement (C) is the incorrect one.
Step 6: Magnetic field check.
For an EM wave $E = vB$, giving $B_y = \dfrac{E_0}{v}\sin(kz - \omega t)$, so statement (D) is correct. The incorrect choice is (C).
\[ \boxed{\text{The wavelength inside the medium is } 300\,\text{m (incorrect)}} \]
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