Question

A plane electromagnetic wave propagating in a non-magnetic dielectric medium is given by $E=E_0\left[4\times 10^{-7}x-50t\right]$, 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,
$E=E_0\left[4\times 10^{-7}x-50t\right]\dots (i)$
General equation of electromagnetic wave
$E=E_0[kx-\omega t]\dots (ii)$
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,
$c=\frac{1}{{\mu }_0{\epsilon }_0}$
$\Rightarrow \frac{v}{c}=\sqrt{\frac{{\mu }_0{\epsilon }_0}{{\mu }_{\text{med }}{\epsilon }_{\text{med }}}}$

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