An electromagnetic wave enters a non-magnetic medium from air. In air, E⃗₁ = E₀₁ x̂ cos[2πν(z/c - t)]. In the medium, E⃗₂ = E₀₂ x̂ cos[k(2z - ct)], where k and ν are the wave number and frequency in air. If εᵣ₁ and εᵣ₂ are the relative permittivities of air and the medium respectively, which relation is correct?
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Question

An electromagnetic wave enters a non-magnetic medium from air. In air, E⃗₁ = E₀₁ x̂ cos[2πν(z/c - t)]. In the medium, E⃗₂ = E₀₂ x̂ cos[k(2z - ct)], where k and ν are the wave number and frequency in air. If εᵣ₁ and εᵣ₂ are the relative permittivities of air and the medium respectively, which relation is correct?

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G1 Physics · Accepted Answer

From the wave in air, E⃗₁ = E₀₁ x̂ cos[2πν(z/c − t)] = E₀₁ x̂ cos[(2πν/c)z − 2πνt]. So the speed in air is c. For the wave in the medium, E⃗₂ = E₀₂ x̂ cos[k(2z − ct)] = E₀₂ x̂ cos[(2k)z − (kc)t]. Therefore the angular frequency in the medium is ω = kc and the wave number is K = 2k. Hence the speed in the medium is v = ω/K = (kc)/(2k) = c/2. For a non-magnetic medium, wave speed is inversely proportional to √εᵣ, so v₁/v₂ = √(εᵣ₂/εᵣ₁). Since v₁ = c and v₂ = c/2, c/(c/2) = 2 = √(εᵣ₂/εᵣ₁). Squaring…