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Question: A plane electromagnetic wave in a non-magnetic dielectric medium is given by\[\bar E = \bar E\left( ...

A plane electromagnetic wave in a non-magnetic dielectric medium is given byEˉ=Eˉ(4×10−7x−50t)\bar E = \bar E\left( {4 \times {{10}^{ - 7}}x - 50t} \right)with distance being in meter and time in seconds. The dielectric constant of the medium is:
A. 5.8
B. 2.4
C. 1.6
D. 3.5

Explanation

Solution

In this question, we need to determine the dielectric constant of the medium whose electro-magnetic wave is defined by Eˉ=Eˉ(4×10−7x−50t)\bar E = \bar E\left( {4 \times {{10}^{ - 7}}x - 50t} \right). For this we will compare the given equation with the general wave equation is given asEˉ=Eˉ(kx−ωt)\bar E = \bar E\left( {kx - \omega t} \right), where ω\omega is the angular frequency and kkis the wave number.

Complete step by step answer:
Eˉ=Eˉ(4×10−7x−50t)−−(i)\bar E = \bar E\left( {4 \times {{10}^{ - 7}}x - 50t} \right) - - (i)
We know the general equation for a wave is given as
Eˉ=Eˉ(kx−ωt)−−(ii)\bar E = \bar E\left( {kx - \omega t} \right) - - (ii)
Now compare the equation (i) and equation (ii), we can write
k=4×10−7m−1k = 4 \times {10^{ - 7}}m{}^{ - 1}
ω=50rads\omega = 50\dfrac{{rad}}{s}
Now we can calculate the velocity of the wave in a nonmagnetic dielectric medium by using the formula v=ωk−−(iii)v = \dfrac{\omega }{k} - - (iii)
Then substitute the values of angular frequency and the wave number in equation (iii), we get
v=ωk=504×10−7=1.25×108msv = \dfrac{\omega }{k} = \dfrac{{50}}{{4 \times {{10}^{ - 7}}}} = 1.25 \times {10^8}\dfrac{m}{s}
We know the dielectric constant in a medium is given by the formula
c=1με0εr−−(iv)c = \dfrac{1}{{\sqrt {\mu {\varepsilon _0}{\varepsilon _r}} }} - - (iv)
This equation can be written as
εr=1με0c2−−(v){\varepsilon _r} = \dfrac{1}{{\mu {\varepsilon _0}{c^2}}} - - (v)
Where
μ=1.25×10−6\mu = 1.25 \times {10^{ - 6}}
ε0=8.85×10−12{\varepsilon _0} = 8.85 \times {10^{ - 12}}
Velocity of the wave v=1.25×108msv = 1.25 \times {10^8}\dfrac{m}{s}
Hence by substituting the values in equation (v) we get

εr=1(1.25×10−6)(8.85×10−12)(1.25×108)2 =1(1.25×10−6)(8.85×10−12)(1.25×108)(1.25×108) =102(1.25)(8.85)(1.25)(1.25) =10217.285 =5.78 ≃5.8  {\varepsilon _r} = \dfrac{1}{{\left( {1.25 \times {{10}^{ - 6}}} \right)\left( {8.85 \times {{10}^{ - 12}}} \right){{\left( {1.25 \times {{10}^8}} \right)}^2}}} \\\ = \dfrac{1}{{\left( {1.25 \times {{10}^{ - 6}}} \right)\left( {8.85 \times {{10}^{ - 12}}} \right)\left( {1.25 \times {{10}^8}} \right)\left( {1.25 \times {{10}^8}} \right)}} \\\ = \dfrac{{{{10}^2}}}{{\left( {1.25} \right)\left( {8.85} \right)\left( {1.25} \right)\left( {1.25} \right)}} \\\ = \dfrac{{{{10}^2}}}{{17.285}} \\\ = 5.78 \\\ \simeq 5.8 \\\

Hence the dielectric constant of the medium is 5.85.8
Option A is correct.

Note: Electromagnetic waves are the waves which can travel through the vacuum of outer space. Electromagnetics are created due to the vibration of the electric charge and this vibration contains both the electric and magnetic components. The speed of propagation of electromagnetic waves in vacuum is 3×108 m/s3 \times {10^8}{\text{ m/s}}.