In this problem, we are given the refractive indices of the ordinary and extraordinary rays for quartz, as well as the wavelength of the incident light. We need to determine the least thickness of the quartz plate for which the ordinary and extraordinary rays combine to form plane polarized light.


Quartz is an anisotropic material, meaning it has different refractive indices for different polarizations of light. This phenomenon is known as birefringence. When a beam of unpolarized light enters a birefringent material, it splits into two rays, the ordinary ray and the extraordinary ray, each with its own refractive index.


To determine the least thickness of the quartz plate, we need to find the condition where the ordinary and extraordinary rays have a phase difference of 180 degrees. This ensures that when they recombine, the resulting light is plane polarized.


The phase difference between the ordinary and extraordinary rays can be calculated using the equation:

Δφ = (2π/λ) * d * (μe - μ0)

Where Δφ is the phase difference, λ is the wavelength of the incident light, d is the thickness of the quartz plate, μe is the refractive index of the extraordinary ray, and μ0 is the refractive index of the ordinary ray.


For the ordinary and extraordinary rays to combine and form plane polarized light, the phase difference must be 180 degrees:

Δφ = 180 degrees


Substituting the given values into the equation, we can solve for the thickness of the quartz plate:

(2π/λ) * d * (μe - μ0) = 180 degrees

d = (180 degrees * λ) / (2π * (μe - μ0))

d = (180 degrees * 5 * 10^-5 cm) / (2π * (1.55 * 0.33 - 1.5442))

d ≈ 0.004 cm


The least thickness of the quartz plate for which the ordinary and extraordinary rays combine to form plane polarized light is approximately 0.004 cm.