For parallel polarisation, for lossless dielectrics, the expression for Brewster angle for a wave traveling from medium 1 to the medium of refractive indices η1 and η2 respectively is

25) For parallel polarisation, for lossless dielectrics, the expression for Brewster angle for a wave traveling from medium 1 to the medium of refractive indices η₁ and η₂ respectively is

• 1. θ = sin^-1 √[ (ε₁/ε₂) / (1 + √(ε₁/ε₂)) ] [Option ID = 2929]
• 2. θ = sin^-1 √[ (ε₂/ε₁) / (1 + ε₂/ε₁) ] [Option ID = 2930]
• 3. θ = sin^-1 √[ ε₂/ε₁ ] [Option ID = 2931]
• 4. θ = cos^-1 √[ ε₁/ε₂ ] [Option ID = 2932]

Answer: B

Solution

To find the expression for the Brewster angle (θ_B) for parallel polarization, follow these steps:

Step 1: Understand the Condition for Brewster's Angle

Brewster's angle occurs when the reflection coefficient for parallel polarization (p-polarization) is zero. For lossless dielectrics, this happens when the reflected ray and the refracted (transmitted) ray are perpendicular (90° apart).

θ_B + θ_t = 90°   ⇒   sin(θ_t) = cos(θ_B)

Step 2: Apply Snell's Law

Using Snell's Law for the interface between medium 1 and medium 2:

n₁ sin(θ_B) = n₂ sin(θ_t)

Substitute the condition from Step 1:

n₁ sin(θ_B) = n₂ cos(θ_B)

Step 3: Relate Refractive Index to Permittivity (ε)

For non-magnetic dielectrics, the refractive index n is related to the relative permittivity ε by n = √ε. Therefore:

tan(θ_B) =

n₂n₁

= √

ε₂ε₁

Step 4: Convert Tangent to Sine

Since the options are in terms of sin^-1, use the identity sin²θ = tan²θ / (1 + tan²θ):

sin²(θ_B) =

(ε₂ / ε₁) 1 + (ε₂ / ε₁)

Step 5: Final Expression

Taking the square root and the inverse sine of both sides gives the standard expression:

θ_B = sin^-1 √[ (ε₂/ε₁) / (1 + ε₂/ε₁) ]

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