Question

A plane electromagnetic wave travelling in a non-magnetic medium is given by $$E=\left(9\times {10}^{8}N{C}^{-1}\right)\sin \left[\left(9\times {10}^{8}rad{s}^{-1}\right)t-\left(6m^{-1}\right)x\right]$$
 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:
$$c={\displaystyle \frac{\omega }{k}}$$

Here, $$\omega$$ is the angular frequency and k is the dielectric constant of the medium.
$$c={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}$$

Here, $$\mu$$ is the permeability of the free space and $$\epsilon$$ 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,$$E=\left(9\times {10}^{8}N{C}^{-1}\right)\sin \left[\left(9\times {10}^{8}rad{s}^{-1}\right)t-\left(6m^{-1}\right)x\right]$$

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 $$k=6m^{-1}$$.

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

$$c=1.5\times {10}^{8}m{s}^{-1}$$


Also, the speed of electromagnetic wave in a dielectric medium of permittivity $$\epsilon$$ is given by the equation,
$$c={\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}$$

Here, $$\mu$$ is the permeability of the medium and $$\epsilon$$ is the permittivity of the medium.

Therefore, we can write,
$${\displaystyle \frac{1}{\sqrt{\mu \epsilon }}}=1.5\times {10}^{8}m{s}^{-1}$$

Squaring the above equation, we get,
$${\displaystyle \frac{1}{\mu \epsilon }}=2.25\times {10}^{16}{m}^2{s}^{-2}$$

$$\Rightarrow \epsilon ={\displaystyle \frac{1}{2.25\times {10}^{16}{m}^2{s}^{-2}\times \mu }}$$

Substitute $$4\pi \times {10}^{-7}Tm{A}^{-1}$$ for $$\mu$$ in the above equation.
$$\Rightarrow \epsilon ={\displaystyle \frac{1}{2.25\times {10}^{16}\times 4\pi \times {10}^{-7}}}$$

$$\epsilon ={\displaystyle \frac{1}{28.26\times {10}^9}}$$

$$\epsilon =0.035\times {10}^{-9}F{m}^{-1}$$

The dielectric constant of the medium is given by,
$$k={\displaystyle \frac{\epsilon }{{\epsilon }_0}}$$

Here, $${\epsilon }_0$$ is the permittivity of the free space and it has value $$8.85\times {10}^{-12}F{m}^{-1}$$.

Substitute $$0.035\times {10}^{-9}F{m}^{-1}$$ for $$\epsilon$$ and $$8.85\times {10}^{-12}F{m}^{-1}$$ for $${\epsilon }_0$$ in the above equation.$$k={\displaystyle \frac{0.035\times {10}^{-9}F{m}^{-1}}{8.85\times {10}^{-12}F{m}^{-1}}}$$

$$k=3.95$$

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 $$1T=kg{s}^{-2}{A}^{-1}$$
 wherever necessary if you don’t remember the units of the above-mentioned parameters.