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Question:

An EM wave from air enters a medium. The electric fields are →E₁=E₀₁^xcos[2πv(z/c−t)] in air and →E₂=E₀₂^xcos[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 εr₁ and εr₂ refer to relative permittivities of air and medium respectively, which of the following options is correct?

εr₁εr₂=1/2

εr₁εr₂=2

εr₁εr₂=4

εr₁εr₂=1/4

Solution:

Velocity of wave v=ω/k
Comparing the given equations with E=Eo cos(kz−ωt), we get
For air : k=2πv/c, ω=2πv ⇒ vair=ω/k=c
For medium : k=2k, ω=kc ⇒ vmedium=ω/k=kc/2k=c/2
Using Snell's law, nair/nmedium=vmedium/vair where ε ∝ n²
∴ εair/εmedium=n²air/n²medium=v²medium/v²air
∴ εair/εmedium=(c/2)²/c²=1/4