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Geometrical Optics

CSCA Geometrical Optics study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Topic: Optics > Geometrical Optics

Geometrical Optics

1. Conceptual Definition

Geometrical optics abstracts light as **"Light Rays"**, ignoring wave details to focus on the propagation path of light. Its core assumptions include:

* **Rectilinear Propagation**: Light travels in straight lines in a homogeneous medium.

* **Independence**: Light rays do not interfere with each other when they cross.

* **Reversibility**: The path of light is reversible (if light travels from A to B, it can travel from B back to A along the same path).

2. Core Laws

#### A. Law of Reflection

The phenomenon where light returns to the original medium upon hitting an interface.

* **Coplanar**: The incident ray, reflected ray, and normal lie in the same plane.

* **Equal Angles**: The angle of reflection equals the angle of incidence ($\theta_r = \theta_i$).

#### B. Law of Refraction (Snell's Law)

The change in direction of light when passing obliquely from one medium to another.

* **Formula**: $$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

* $n_1, n_2$: Absolute refractive indices of the first and second media.

* $\theta_1, \theta_2$: Angle of incidence and angle of refraction (**angles relative to the normal**).

* **Refractive Index**: $$n = \frac{c}{v}$$ (where $c$ is speed of light in vacuum, $v$ is speed in the medium).

#### C. Total Internal Reflection (TIR)

Occurs when light travels from an **optically denser medium** (higher $n$) to an **optically rarer medium** (lower $n$), and the angle of incidence $\theta_i$ exceeds the **Critical Angle $\theta_c$**. All light is reflected back.

* **Critical Angle Formula**: $$\sin\theta_c = \frac{n_{small}}{n_{large}}$$

3. Problem Solving Strategy

1. **Draw**: Always sketch the ray diagram, labeling the interface and the **Normal**.

2. **Identify Angles**: Determine angles of incidence and refraction, ensuring they are measured **relative to the normal**, not the surface.

3. **Calculate**: Apply Snell's Law or geometric relationships.

4. Typical Examples

**Example 1: Law of Reflection (Rotating Mirror)**

A light ray strikes a plane mirror perpendicularly. If the incident ray is fixed and the mirror rotates by an angle $\alpha$, by how much does the reflected ray rotate?

* **Analysis**: The normal rotates by $\alpha$, making the new angle of incidence $\alpha$. By the law of reflection, the angle of reflection is also $\alpha$. The total angle between incident and reflected rays becomes $2\alpha$. Thus, the reflected ray rotates by **$2\alpha$**.

**Example 2: Refraction and TIR**

Light travels from glass ($n=1.5$) to air ($n \approx 1$).

* (1) If the angle of incidence is $30^\circ$, find the angle of refraction.

* Solution: $1.5 \times \sin 30^\circ = 1.0 \times \sin \theta_2 \Rightarrow \sin \theta_2 = 0.75 \Rightarrow \theta_2 \approx 48.6^\circ$.

* (2) Find the critical angle for total internal reflection.

* Solution: $\sin \theta_c = 1.0 / 1.5 = 2/3 \Rightarrow \theta_c \approx 41.8^\circ$.