Problem 5.85 — Polarization of Rayleigh Scattered Light

Problem Statement

Sunlight travelling in the $x$-direction is scattered by air molecules. Find the degree of polarization of light scattered at $90°$ to the incident beam (i.e., in the $y$-direction).

Given Information

• See problem statement for all given quantities.

Physical Concepts & Formulas

This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.

• See the step-by-step solution for the specific equations applied.
• All quantities are in SI units unless otherwise stated.

Step-by-Step Solution

Step 1 — Identify given quantities and set up the problem: For Rayleigh scattering, the scattered intensity for the component parallel to the scattering plane is zero at 90° (since $\cos^2 90° = 0$). Only the perpendicular component survives.

Step 2 — Apply the relevant physical law or equation: $$P = \frac{I_\perp – I_\parallel}{I_\perp + I_\parallel} = \frac{I_\perp – 0}{I_\perp + 0} = \boxed{1.0 \text{ (100% polarized)}}$$

Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.

Worked Calculation

$$P = \frac{I_\perp – I_\parallel}{I_\perp + I_\parallel} = \frac{I_\perp – 0}{I_\perp + 0} = \boxed{1.0 \text{ (100% polarized)}}$$

$$\text{Numerical result} = \text{given expression substituted with values}$$

$$\boxed{P = \frac{I_\perp – I_\parallel}{I_\perp + I_\parallel} = \frac{I_\perp – 0}{I_\perp + 0} = \boxed{1.0 \text{ (100% polarized)}}}$$

For Rayleigh scattering, the scattered intensity for the component parallel to the scattering plane is zero at 90° (since $\cos^2 90° = 0$). Only the perpendicular component survives.

$$P = \frac{I_\perp – I_\parallel}{I_\perp + I_\parallel} = \frac{I_\perp – 0}{I_\perp + 0} = \boxed{1.0 \text{ (100% polarized)}}$$

Answer

$$P = \frac{I_\perp – I_\parallel}{I_\perp + I_\parallel} = \frac{I_\perp – 0}{I_\perp + 0} = \boxed{1.0 \text{ (100% polarized)}}$$

Physical Interpretation

The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.