Problem 5.128 — Scattering: Angular Distribution

Problem Statement

Rayleigh scattering of sunlight. Find the ratio of scattered intensity at $\theta = 90°$ to that at $\theta = 0°$ for a single dipole oscillator.

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 a dipole oscillator, the scattered intensity pattern is $I \propto \sin^2\psi$ where $\psi$ is the angle from the dipole axis. For an oscillator driven by $x$-polarized light, at $\theta = 90°$ in the forward plane: $I(90°)/I(0°) = \sin^2(90°)/\sin^2(0°)$ which diverges. More carefully:

Step 2 — Apply the relevant physical law or equation: For unpolarized incident light, $I(\theta)/I(0) = \frac{1+\cos^2\theta}{2}$. At $\theta = 90°$: $I = (1+0)/2 = 0.5$. At $\theta = 0°$: $I = (1+1)/2 = 1$.

Step 3 — Solve algebraically for the unknown: $$\boxed{I(90°)/I(0°) = 0.5}$$

Worked Calculation

$$\boxed{I(90°)/I(0°) = 0.5}$$

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

$$\boxed{\boxed{I(90°)/I(0°) = 0.5}}$$

For a dipole oscillator, the scattered intensity pattern is $I \propto \sin^2\psi$ where $\psi$ is the angle from the dipole axis. For an oscillator driven by $x$-polarized light, at $\theta = 90°$ in the forward plane: $I(90°)/I(0°) = \sin^2(90°)/\sin^2(0°)$ which diverges. More carefully:

For unpolarized incident light, $I(\theta)/I(0) = \frac{1+\cos^2\theta}{2}$. At $\theta = 90°$: $I = (1+0)/2 = 0.5$. At $\theta = 0°$: $I = (1+1)/2 = 1$.

$$\boxed{I(90°)/I(0°) = 0.5}$$

Answer

$$\boxed{I(90°)/I(0°) = 0.5}$$

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.