Question

A plane electromagnetic wave in a non-magnetic dielectric medium is given by $\overrightarrow{E}=\overrightarrow{{{E}_{0}}}\sin (4\times {{10}^{-7}}x-50t)$ with distance being in meter and time in seconds. The dielectric constant of the medium is:
$\begin{aligned} & (A)2.4 \\\ & (B)5.8 \\\ & (C)8.2 \\\ & (D)4.8 \\\ \end{aligned}$

Answer 1

We will use the given wave equation and compare it with a standard wave equation. This will give us the value of wave number (k) and angular frequency ($\omega$). Now, the wave velocity is given by angular frequency upon wave number. Once, we get the wave velocity, we can calculate the refractive index of the medium and then finally calculate the dielectric constant of the medium.

Complete answer:
The wave equation given to us is:
$\Rightarrow \overrightarrow{E}=\overrightarrow{{{E}_{0}}}\sin (4\times {{10}^{-7}}x-50t)$
On comparing it with standard wave equation, that is:
$\Rightarrow \overrightarrow{E}=\overrightarrow{{{E}_{0}}}\sin (kx-\omega t)$
We get,
The value of wave number (k) as: $4\times {{10}^{-7}}{{m}^{-1}}$
And the value of angular frequency as: $50{{s}^{-1}}$
Thus, the velocity of wave (say v) can be calculated as:
$\begin{aligned} & \Rightarrow v=\dfrac{\omega }{k} \\\ & \Rightarrow v=\dfrac{50}{4\times {{10}^{-7}}} \\\ & \Rightarrow v=1.25\times {{10}^{8}}m{{s}^{-1}} \\\ \end{aligned}$
Once, we have the velocity of the wave. We can calculate the refractive index of the di-electric medium as follows:
$\Rightarrow \mu =\dfrac{c}{v}$
Where, c is the speed of light.
Putting the values of all the terms in the above equation, we get:
$\begin{aligned} & \Rightarrow \mu =\dfrac{3\times {{10}^{8}}}{1.25\times {{10}^{8}}} \\\ & \Rightarrow \mu =2.4 \\\ \end{aligned}$
Now, the relation between the refractive index of a medium and its magnetic and di-electric constant is given by:
$\Rightarrow {{\mu }^{2}}={{K}_{m}}\times {{K}_{d}}$
Since, the medium is given to be non-magnetic, therefore:
$\Rightarrow {{K}_{m}}=1$
Using this in above equation and putting the value of refractive index, we get:
$\begin{aligned} & \Rightarrow {{K}_{d}}={{(2.4)}^{2}} \\\ & \Rightarrow {{K}_{d}}=5.76 \\\ & \therefore {{K}_{d}}\approx 5.8 \\\ \end{aligned}$
Hence, the dielectric constant of the medium comes out to be 5.8 .

Hence, option (B) is the correct option.

Note:
We should always remember the common values like magnetic constant of magnetic, non-magnetic, dia-magnetic substances. Also, in problems like these we should proceed in a step by step manner and keep in check of our solution at every step. As even one simple error can result in the wrong solution and answer.