Calculate The Reflectance of A Sheet of Polystyrene N 1.59

Polystyrene is a common plastic material with a refractive index of approximately 1.59. Calculating its reflectance helps in understanding how much light is reflected from a sheet of polystyrene at different angles of incidence. This calculation is important in optical design, packaging, and material science applications.

How to calculate the reflectance of polystyrene

The reflectance of a material depends on several factors including the angle of incidence, the refractive index of the material, and the refractive index of the surrounding medium (typically air, which has n=1). The Fresnel equations are used to calculate the reflectance for both parallel (s-polarized) and perpendicular (p-polarized) light waves.

Formula used

The reflectance (R) for s-polarized light is calculated as:

R_s = (n₁cosθ_i - n₂cosθ_t) / (n₁cosθ_i + n₂cosθ_t)

The reflectance for p-polarized light is calculated as:

R_p = (n₂cosθ_i - n₁cosθ_t) / (n₂cosθ_i + n₁cosθ_t)

Where:

n₁ = refractive index of the surrounding medium (air, n=1)
n₂ = refractive index of polystyrene (n=1.59)
θ_i = angle of incidence (in degrees)
θ_t = angle of refraction (in degrees)

The total reflectance is the average of R_s and R_p.

Steps to calculate

Determine the angle of incidence (θ_i) in degrees.
Calculate the angle of refraction (θ_t) using Snell's law: n₁sinθ_i = n₂sinθ_t.
Calculate R_s using the first formula above.
Calculate R_p using the second formula above.
Calculate the total reflectance as (R_s + R_p)/2.

Note: The angle of incidence must be between 0° and 90°. For angles greater than the critical angle, total internal reflection occurs, and the reflectance is 100%.

Worked example

Let's calculate the reflectance of a sheet of polystyrene (n=1.59) when light strikes it at a 30° angle.

Step 1: Calculate the angle of refraction

Using Snell's law:

1 × sin(30°) = 1.59 × sin(θ_t)

0.5 = 1.59 × sin(θ_t)

sin(θ_t) = 0.5 / 1.59 ≈ 0.3144

θ_t ≈ arcsin(0.3144) ≈ 18.28°

Step 2: Calculate R_s

R_s = (1 × cos(30°) - 1.59 × cos(18.28°)) / (1 × cos(30°) + 1.59 × cos(18.28°))

R_s ≈ (0.8660 - 1.59 × 0.9487) / (0.8660 + 1.59 × 0.9487)

R_s ≈ (0.8660 - 1.5196) / (0.8660 + 1.5196)

R_s ≈ (-0.6536) / (2.3856) ≈ -0.2740

Since reflectance cannot be negative, we take the absolute value: R_s ≈ 0.2740 or 27.40%

Step 3: Calculate R_p

R_p = (1.59 × cos(30°) - 1 × cos(18.28°)) / (1.59 × cos(30°) + 1 × cos(18.28°))

R_p ≈ (1.59 × 0.8660 - 0.9487) / (1.59 × 0.8660 + 0.9487)

R_p ≈ (1.3846 - 0.9487) / (1.3846 + 0.9487)

R_p ≈ 0.4359 / 2.3333 ≈ 0.1868 or 18.68%

Step 4: Calculate total reflectance

Total reflectance = (R_s + R_p) / 2 = (0.2740 + 0.1868) / 2 ≈ 0.2304 or 23.04%

Therefore, at a 30° angle of incidence, approximately 23.04% of light is reflected from a sheet of polystyrene.

FAQ

What is the refractive index of polystyrene?

The refractive index of polystyrene is approximately 1.59, which means light travels 1.59 times slower in polystyrene than in air.

How does the angle of incidence affect reflectance?

The reflectance increases as the angle of incidence approaches 90°. At normal incidence (0°), the reflectance is at its minimum, and at grazing angles (near 90°), the reflectance approaches 100%.

What is the difference between s-polarized and p-polarized reflectance?

S-polarized light has its electric field perpendicular to the plane of incidence, while p-polarized light has its electric field parallel to the plane of incidence. The reflectance for these two polarizations differs, and the total reflectance is the average of the two.

Can I use this calculator for other materials?

Yes, you can modify the calculator to use different refractive indices for other materials. The same formulas apply as long as you know the refractive index of the material.