An EM wave from air enters a medium. The electric fields are $\overrightarrow{\mathrm{E}}_{1}=\mathrm{E}_{01} \hat{x} \cos \left[2 \pi \nu\left(\frac{z}{c}-\mathrm{t}\right)\right]$ in air and $\overrightarrow{\mathrm{E}}_{2}=\mathrm{E}_{02} \hat{x} \cos [\mathrm{k}(2 z-\mathrm{ct})]$ in medium, where the wave number k and frequency $\nu$ refer to their values in air. The medium is non-magnetic. If $\epsilon_{\mathrm{r}_{1}}$ and $\epsilon_{\mathrm{r}_{2}}$ refer to relative permittivities of air and medium respectively, which of the following options is correct ?

(D) $\frac{\epsilon_{\mathrm{r}_{1}}}{\epsilon_{\mathrm{r}_{2}}}=\frac{1}{4}$

(A) $\frac{\epsilon_{r_{1}}}{\epsilon_{r_{2}}}=\frac{1}{2}$

(B) $\frac{\epsilon_{\mathrm{r}_{1}}}{\epsilon_{\mathrm{r}_{2}}}=4$

(C) $\frac{\epsilon_{r_{1}}}{\epsilon_{r_{2}}}=2$

JEE Main · 2018 · 08 Apr 2018 · Offline · EM Waves

An EM wave from air enters a medium. The electric fields are

E_{1} = E_{01} \overset{x}{^} cos [2 π ν (\frac{z}{c} - t)]

in air and

E_{2} = E_{02} \overset{x}{^} cos [k (2 z - ct)]

in medium, where the wave number k and frequency

ν

refer to their values in air. The medium is non-magnetic. If

ϵ_{r_{1}}

and

ϵ_{r_{2}}

refer to relative permittivities of air and medium respectively, which of the following options is correct ?

Answer: (D)

Reading the wave speeds off the two forms, the wave travels at

c

in air, while in the medium

cos [k (2 z - c t)] = cos [2 k (z - \frac{c}{2} t)]

shows a speed of

\frac{c}{2}

. The refractive index is the ratio of these speeds,

\frac{c}{c /2} = 2

, and for a non-magnetic medium the index is

ϵ_{r}

, so

\frac{ϵ _{r 2}}{ϵ _{r 1}} = 2^{2} = 4

, which means

\frac{ϵ _{r 1}}{ϵ _{r 2}} = \frac{1}{4}

JEE Main · 2018 · 08 Apr 2018 · Offline · Single correct option