An EM wave from air enters a medium. The electric fields are
`vec(E )_(1)=E_(01)hat(x)cos[2pi v((z)/(c )-t)]` in air and `vec(E )_(2)=E_(02)hat(x)cos [k(2z-ct)]` in medium, where the wave number k and frequency v refer to their values in air. The medium is non - magnetic. If `in_(r_(1))` and `in_(r_(2))` refer to relative permittivities of air and medium respectively, which of the following options is correct ?

Options

A. `(in_(r_(1)))/(in_(r_(2)))=4`
B. `(in_(r_(1)))/(in_(r_(2)))=2`
C. `(in_(r_(1)))/(in_(r_(2)))=(1)/(4)`
D. `(in_(r_(1)))/(in_(r_(2)))=(1)/(2)`

Answer

C

Explanation

`vec(E )_(1)=E_(01)hat(x)cos [2pi f((Z)/(c )-t)]`
`vec(E )_(2)=_(02)hat(x)cos [k(2z-(ct)]`
Equation (1), `vec(E )_(1)=E_(01)hat(x) cos [(2pi c)/(lambda)((z)/(c )-t)]`
`=E_(01)hat(x)cos[kc ((z)/(c )-t)]`
`= E_(01)hat(x)cos [k (z-ct)] " "` ....(1)
and `vec(E )_(2)=E_(02)hat(x)cos [K (2Z-ct)]`
`= E_(02) hat(x) cos [2k(Z-(c )/(2)t)] " "` ....(2)
From equation (1), `k_(1)=k`, From equation (2),
`k_(2)=2k`
but `k = sqrt(in_(4))`
`((k_(1))/(k_(2)))^(2)=((in_(r_(1)))/(in_(r_(2))))`
`therefore ((k)/(2k))^(2)=(in_(r_(1)))/(in_(r_(2)))`
`therefore ((1)/(2))^(2)=(in_(r_(1)))/(in_(r_(2))) " " therefore (in_(r_(1)))/(in_(r_(2)))=(1)/(4)`