Question

A plane electromagnetic wave travelling in a non-magnetic medium is given by \[
E = \left( {9 \times 10^8 \,NC^{ - 1} } \right)\sin \left[ {\left( {9 \times 10^8 \,rad\,s^{ - 1} } \right)t - \left( {6\,m^{ - 1} } \right)x} \right]
\]
 where x is in meters and t is in second. What will be the dielectric constant of the medium?
A. 5
B. 4
C. 3
D. 2

Answer 1

Hint: compare the given equation of electric field with the standard equation of electric field of the electromagnetic wave.

Formula used:
\[c = \dfrac{\omega }{k}\]

Here, \[\omega \] is the angular frequency and k is the dielectric constant of the medium.
\[c = \dfrac{1}{{\sqrt {\mu \varepsilon } }}\]

Here, \[\mu \] is the permeability of the free space and \[\varepsilon \] is the dielectric constant of the air.

Complete step by step answer:
The electric field of the electromagnetic wave is given by the equation,\[E = E_0 \sin \left( {\omega t - kx} \right)\] ...... (1)

Here, \[E_0 \] is the initial electric field, \[\omega \] is the angular frequency, t is the time, k is the wave number, and x is the distance.

The given equation of the electric field is,\[E = \left( {9 \times 10^8 \,NC^{ - 1} } \right)\sin \left[ {\left( {9 \times 10^8 \,rad\,s^{ - 1} } \right)t - \left( {6\,m^{ - 1} } \right)x} \right]\]

Compare the above equation with the standard equation of electric field of the electromagnetic wave (1). We get,
\[\omega = 9 \times 10^8 \,rad\,s^{ - 1} \]and \[k = 6\,m^{ - 1} \].

The speed of propagation of the wave is given by,
\[c = \dfrac{\omega }{k}\]
Substitute \[9 \times 10^8 \,rad\,s^{ - 1} \] for \[\omega \] and \[6\,m^{ - 1} \] for \[k\] in the above equation.
\[c = \dfrac{{9 \times 10^8 \,rad\,s^{ - 1} }}{{6\,m^{ - 1} }}\]

\[c = 1.5 \times 10^8 \,ms^{ - 1} \]


Also, the speed of electromagnetic wave in a dielectric medium of permittivity \[\varepsilon \] is given by the equation,
\[c = \dfrac{1}{{\sqrt {\mu \varepsilon } }}\]

Here, \[\mu \] is the permeability of the medium and \[\varepsilon \] is the permittivity of the medium.

Therefore, we can write,
\[\dfrac{1}{{\sqrt {\mu \varepsilon } }} = 1.5 \times 10^8 \,ms^{ - 1} \]

Squaring the above equation, we get,
\[\dfrac{1}{{\mu \varepsilon }} = 2.25 \times 10^{16} \,m^2 s^{ - 2} \]

\[ \Rightarrow \varepsilon = \dfrac{1}{{2.25 \times 10^{16} \,m^2 s^{ - 2} \times \mu }}\]

Substitute \[4\pi \times 10^{ - 7} \,TmA^{ - 1} \] for \[\mu \] in the above equation.
\[ \Rightarrow \varepsilon = \dfrac{1}{{2.25 \times 10^{16} \times 4\pi \times 10^{ - 7} }}\]

\[\varepsilon = \dfrac{1}{{28.26 \times 10^9 }}\]

\[\varepsilon = 0.035 \times 10^{ - 9} \,Fm^{ - 1} \]

The dielectric constant of the medium is given by,
\[k = \dfrac{\varepsilon }{{\varepsilon _0 }}\]

Here, \[\varepsilon _0 \] is the permittivity of the free space and it has value \[8.85 \times 10^{ - 12} \,Fm^{ - 1} \].

Substitute \[0.035 \times 10^{ - 9} \,Fm^{ - 1} \] for \[\varepsilon \] and \[8.85 \times 10^{ - 12} \,Fm^{ - 1} \] for \[\varepsilon _0 \] in the above equation.\[k = \dfrac{{0.035 \times 10^{ - 9} \,Fm^{ - 1} }}{{8.85 \times 10^{ - 12} \,Fm^{ - 1} }}\]

\[k = 3.95\]

Therefore, the dielectric constant is nearly 4.

So, the correct answer is “Option B”.
Note:
Remember, the units of permittivity of the medium and permeability of the medium are in S.I. units. Also use \[1\,T = kg\,s^{ - 2} A^{ - 1} \]
 wherever necessary if you don’t remember the units of the above-mentioned parameters.