A plane electromagnetic wave propagating in a non-magnetic dielectric medium is given by `E=E_(0)[4 times 10^(-7)x-50t]`, 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

Options

A. 2.42
B. 5.76
C. 8.26
D. 4.84

Answer

B

Explanation

Here, equation of the electromagnetic wave,
`" "E=E_(0)[4 times 10^(-7)x-50t] " …(i)"`
General equation of electromagnetic wave
`" "E=E_(0)[kx-omegat] " …(ii)"`
So, by the compairing Eqs. (i) and (ii), we get
`" "k=4 times 10^(-7) " and "omega=50`
Hence, wavelength, `lambda=(2pi)/(4 times 10^(-7))m " "(because k=(2pi)/lambda)`
and frequency, `f=50/(2pi)Hz " "(because omega=2pif)`
`therefore` Speed of light, `v=flambda`
`rArr " "v=50/(2pi) times (2pi)/(4 times 10^(-7))=12.5 times 10^(7)m//s`
As, we know the speed of light,
`" "c=1/(mu_(0)epsilon_(0))`
`rArr " "v/c=sqrt((mu_(0)epsilon_(0))/(mu_("med")epsilon_("med")))`
`rArr " "mu_(r)epsilon_(r)=c^(2)/v^(2)[{:(thereforemu_(r)=mu_(med)/mu_(0) " and "), (epsilon_(r)=epsilon_(med)/epsilon_(0)):}`
`rArr " "epsilon_(r)=c^(2)/v^(2) " "(because mu_(r)=1)`
`rArr " "epsilon_(r)=((3 times 10^(8))/(12.5 times 10^(7)))^(2)=5.76`
Hence, the dielectric constant of the medium is 5.76.