Question

A plane electromagnetic wave propagating in a non-magnetic dielectric medium is given by $E=E_{0}\left[4 \times 10^{-7}\,x-50\, t\right]$, where $x$ is in metre and $t$ is in second. If the relative permeability of the medium, $\mu_{r}=1$ then the dielectric constant of the medium is

• A. 2.42
• B. 5.76
• C. 8.26
• D. 4.84

Answer 1

Answer: (B)

Here, equation of the electromagnetic wave,
$E=E_{0}\left[4 \times 10^{-7} x-50 t\right]\,\,\,\ldots(i)$
General equation of electromagnetic wave
$E=E_{0}[k x-\omega t]\,\,\,\ldots(ii)$
So, by the compairing Eqs. (i) and (ii), we get
$k=4 \times 10^{-7} $ and $\omega=50$
Hence, wavelength, $\lambda=\frac{2 \pi}{4 \times 10^{-7}} m \,\,\,\left(\because k=\frac{2 \pi}{\lambda}\right)$
and frequency, $f=\frac{50}{2 \pi} H\,\,\,(\because \omega=2 \pi f)$
$\therefore $ Speed of light, $v=f \lambda$
$\Rightarrow v=\frac{50}{2 \pi} \times \frac{2 \pi}{4 \times 10^{-7}}=12.5 \times 10^{7}\, m /\, s$
As, we know the speed of light,
$c=\frac{1}{\mu_{0} \varepsilon_{0}}$
$\Rightarrow \frac{v}{c}=\sqrt{\frac{\mu_{0} \varepsilon_{0}}{\mu_{\text {med }} \varepsilon_{\text {med }}}}$

$\Rightarrow \varepsilon_{r}=\frac{c^{2}}{v^{2}} \,\,\,\left(\because \mu_{r}=1\right)$
$\Rightarrow \varepsilon_{r}=\left(\frac{3 \times 10^{8}}{12.5 \times 10^{7}}\right)^{2}=5.76$
Hence, the dielectric constant of the medium is $5.76$.