Brewster angle for parallel (p-) polarization: air (n1 = 1) to paraffin with εr = 3 (n2 = √3). Find θB.

Introduction / Context:
The Brewster angle is the angle of incidence at which the reflected component of a parallel (p-) polarized wave is zero at a dielectric interface. It is a fundamental optics and electromagnetics concept used in polarization control and anti-reflection design.

Given Data / Assumptions:

• Incidence from air (n1 = 1) onto paraffin with relative permittivity εr = 3.
• Refractive index of paraffin: n2 = √εr = √3 ≈ 1.732 (assuming μr = 1).
• Parallel polarization (p-pol) and non-magnetic media.

Concept / Approach:

Brewster angle for p-polarization is θB = arctan(n2/n1). With air to dielectric, θB depends only on the refractive index ratio (if μr ≈ 1).

Step-by-Step Solution:

Verification / Alternative check:

Snell’s law and Fresnel coefficients confirm that for p-polarization, the reflection coefficient goes to zero at θB where transmitted and reflected rays are orthogonal, which occurs at arctan(n2/n1).

Why Other Options Are Wrong:

• 0°, 30°, 45°: do not satisfy arctan(√3) ≈ 60°.
• 75°: too large for n2/n1 = √3; arctan growth is slower than linear.

Common Pitfalls:

• Using s-polarization formula (no Brewster zero for s-pol at dielectric-dielectric).
• Using θB = arctan(n1/n2) (inverted ratio) which would be incorrect for incidence from air to denser medium.

Final Answer:

60°